Numerical Investigation of Transient Flow Responses in ... · industry to shift their focus towards...

$: Numerical Investigation of Transient Flow Responses in ... · industry to shift their focus towards the low permeability reservoirs such as shale or tight rock reservoir. Multi-fractured$
Numerical Investigation of Transient Flow Responses in Fractured Tight Oil Wells

by

Min Yue

A thesis submitted in partial fulfillment of the requirements for the degree of

Master of Science

in

Petroleum Engineering

Department of Civil and Environmental Engineering

University of Alberta

© Min Yue, 2016

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Abstract

Increasing demand of global energy and limited conventional resources force the petroleum

industry to shift their focus towards the low permeability reservoirs such as shale or tight rock

reservoir. Multi-fractured horizontal wells have economically unlocked the massive hydrocarbon

resources from unconventional reservoirs. Horizontal drilling and hydraulic fracturing create a

complex fracture network that could enhance reservoir contact area to achieve economic

production rates.

In this study, we compute the transient response in a segment of a hydraulically fractured

horizontal well using a triple-porosity model. Impacts of capillary discontinuity (fracture face-

effect) and some limitations in analytical models such as sequential flow, single-phase flow and

fully-connected symmetric fractures are investigated.

We find that the uncertainty in model history-matched parameters and assumptions associated

with analytical models could potentially over- or under-estimate production by up to 30%.

History-matching with analytical models alone and the assumption of uniformly-spaced fracture

stages would tend to overestimate long-term production forecast. In contrast, the assumption of

no solution gas in tight oil reservoir leads to underestimation of reservoir properties such as

length of fracture and permeability. Moreover, the simulated production data indicates that

fracture face-effect results in rapid production decline. Lower capillary contrast between fracture

and matrix results in less water blockage and higher production.

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Dedicated to my parents, grandparents and fiancé, who give me courage, support and endless love

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Acknowledgments

First, I would like to express my sincerest gratitude to my supervisors, Dr. Juliana Y. Leung and

Dr. Hassan Dehghanpour, for supporting me, for their constant encouragement, guidance and

help throughout my project with their patience.

I would like to thank Dr. Alireza Nouri and Dr. Yuntong She, my examining committee

members.

I am thankful to support of Natural Sciences and Engineering Research Council of Canada

(NSERC). Also I am thankful Computer Modelling Group Ltd. (CMG) for providing academic

licenses of IMEX for my research.

I would like to thank Mingyuan who has given me his valuable suggestions in model building. I

would like to extend my thanks to my dearest friends Yang, Xinming, Bo, Jingwen, Zhongguo,

Cui, Song, Jindong and Yanming for their warm friendship.

Last but not the least, I would like to express my love to my family: my parents, my grandparents

and Mingxiang, who have been with me through the best and worst times with their endless love

and support.

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Table of Content

1 General Introduction ....................................................................................................... 1

1.1. Overview and Background ............................................................................................ 1

1.2. Problem Statement ......................................................................................................... 3

1.3. Research Objectives ....................................................................................................... 4

1.4. Thesis Layout .................................................................................................................. 6

2 Literature Review ............................................................................................................. 7

2.1 A Brief Background of Unconventional Reservoirs .................................................... 7

2.2 Modeling and Analysis of Flow Response in Hydraulically-Stimulated Tight

Reservoirs .................................................................................................................................. 8

2.3 Modeling and Analysis of Multiphase Flow Response in Tight Reservoirs ............ 10

3 Geological and Production Characteristics of Tight Oil Reservoirs ............ 12

3.1 Geological Characteristics ........................................................................................... 13

3.2 Hydraulic Fracturing ................................................................................................... 16

3.3 Natural Fracture (Secondary Fracture) ..................................................................... 20

4 Methodology ..................................................................................................................... 25

4.1 Rate Transient Analysis ............................................................................................... 26

4.2 Analytical Solution: Using Laplace Transform ......................................................... 27

VI

4.3 History Matching and Sensitivity Analysis of Simulation Models .......................... 31

5 Numerical Investigation of Limitations and Assumptions of Analytical

Transient Flow Models ........................................................................................................ 32

5.1 Configuration of Multistage Hydrualic Fracturing .................................................. 34

5.2 Intensity/Spacing of SecondaryFractures .................................................................. 37

5.3 Non-Sequential Flow .................................................................................................... 42

5.4 Non-Symmetric Drainage ............................................................................................ 46

5.5 Heterogeneous Fracture Properties ............................................................................ 54

6 Influence of Multi-phase Effects on oil Production ............................................ 60

6.1 The Role of Gas Dissolution ........................................................................................ 60

6.2 The Role of Capillary Discontinuity ........................................................................... 66

7 Conclusions and Future Work ................................................................................... 90

7.1 Conclusions ................................................................................................................... 90

7.2 Future Work ................................................................................................................. 93

References ................................................................................................................................. 94

VII

List of Tables

Table 5.1 Known Field Data ...................................................................................................... 32

Table 5.2 Unknown Data – Assumed based on typical values observed in Bakken reservoirs

....................................................................................................................................................... 33

Table 5.3 Unknown Data – Estimated based on rate transient analysis (RTA) .................... 33

Table 5.4 Spacing between the four stages of hydraulic fractures for the Cardium well in

Case A .......................................................................................................................................... 34

Table 5.5 History-matched hydraulic fracture half-length and the corresponding number

of natural fractures for the Bakken well ................................................................................... 37

Table 5.6 Spacing between natural fractures for the Bakken well in Case A ....................... 46

Table 5.7 Natural fracture length for the Bakken well in Case B .......................................... 50

Table 6.1 Water saturation at interface in dual-porosity and triple-porosity models with

different capillary pressure (Pc1 and Pc3) ................................................................................. 78

Table 6.2 Value of rowk in conditions of different relative permeability curvature ........... 88

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List of Figures

Figure 3.1 Schematic north-south cross section showing the Bakken and adjacent

formations (USGS, 2013) ............................................................................................................ 13

Figure 3.2 Proppant distribution in hydraulic fracture (Cipolla et al. 2009) ........................ 17

Figure 3.3 Horizontal well with transversely intersecting fracture (Phillips, et al., 2007) ... 19

Figure 4.1 Top view of a horizontal well in triple-porosity model ......................................... 25

Figure 4.2 Analysis of the Cardium well production data with analytical models ............... 30

Figure 5.1 Pressure (in kPa) distribution at the end of production history (250 days) for the

Cardium well with different distribution of LF ........................................................................ 35

Figure 5.2 Log-log plot of daily oil rate as a function of time for the Cardium well in Case A

over 250 days ............................................................................................................................... 36

Figure 5.3 Relationship between xf –nf for the Bakken well ................................................... 38

Figure 5.4 Schematic of the base case with eight natural fractures for the Bakken well ..... 40

Figure 5.5 Pressure (in kPa) distribution for the base case (Bakken well) at the end of

production history (4.5 years) .................................................................................................... 40

Figure 5.6 Log-log plot of daily oil rate as a function of time for the Bakken well .............. 41

Figure 5.7 Pressure (in kPa) distribution for the sequential case (Bakken well) at the end of

production history (4.5 years) .................................................................................................... 42

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Figure 5.8 Log-log plot of daily oil rate as a function of time for the base case and the

sequential case (Bakken well) .................................................................................................... 43

Figure 5.9 Log-log plot of daily oil rate as a function of time for the base case and all

sequential cases (Bakken well) ................................................................................................... 44

Figure 5.10 Log-log plot of daily oil rate as a function of time for non-sequential and

sequential cases............................................................................................................................ 45

Figure 5.11 Pressure (in kPa) distribution of the Bakken well at 4.5 years and the

histogram of Lf for ...................................................................................................................... 47

Figure 5.12 Log-log plot of daily oil rate as a function of time of the Bakken well for Case A

....................................................................................................................................................... 48


with higher matrix permeability (0.0026mD) ........................................................................... 49

Figure 5.14 Schematic of non-symmetric drainage for the Bakken well ............................... 51

Figure 5.15 Pressure (in kPa) distribution of the Bakken well after 4.5 years for ............... 52

Figure 5.16 Log-log plot of daily oil rate as a function of time of the Bakken well for Case B

....................................................................................................................................................... 54

Figure 5.17 Schematic of fully-conncected and non-conncected natural fractures for the

Bakken well.................................................................................................................................. 55

Figure 5.18 Log-log plot of daily oil rate as a function of time of the Bakken well for cases

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with varying natural fracture connectivity (4.5 years) ............................................................ 56

Figure 5.19 Schematic of fully-conncected and non-conncected hydraulic fractures for the

Bakken well.................................................................................................................................. 57

Figure 5.20 Log-log plot of daily oil rate as a function of time of the Bakken well for cases

with varying hydraulic fracture connectivity (4.5years) ......................................................... 58

Figure 5.21 Pressure (in kPa) distribution of the Bakken well after 4.5 years when

RL=0.224 ....................................................................................................................................... 59

Figure 6.1 Average gas saturation in hydraulic fracture during the first 30 days for the

Bakken well.................................................................................................................................. 61

Figure 6.2 Input PVT data for the Bakken models with different bubble-point pressure .. 62

Figure 6.3 Log-log plots of production profiles for the Bakken models with different

bubble-point pressure ................................................................................................................. 63


relative permeability endpoints ................................................................................................. 64


relative permeability curvatures ............................................................................................... 65

Figure 6.6 Oil and water saturation profiles in matrix as a function of distance from the

fracture-matrix interface for the dual-porosity model at: ...................................................... 68

Figure 6.7 Different capillary pressure curve for matrix ........................................................ 69

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Figure 6.8 Oil and water saturation profiles in matrix as a function of time for the dual-

porosity model with different capillary pressure:.................................................................... 70

Figure 6.9 Oil and water saturation profiles in saturation in matrix as a function of

distance from the fracture-matrix interface for the dual-porosity models with different

capillary pressure at 30 days: .................................................................................................... 71

Figure 6.10 Log-log plot of daily rate versus time for the dual-porosity models with

different capillary pressure in matrix during 91 days: ........................................................... 72

Figure 6.11 Average oil saturation in fracture in models in as a function of time for the

dual-porosity model different capillary pressure: ................................................................... 73

Figure 6.12 Log-log plot of cumulative oil versus time for the dual-porosity models with

different capillary pressure in matrix during 91 days ............................................................. 74



Figure 6.14 Oil and water saturation profiles in matrix as a function of .............................. 77

Figure 6.15 Triple-porosity models with different capillary pressure: .................................. 78

Figure 6.16 Different relative permeability in matrix system by changing endpoint of krow80

Figure 6.17 Saturation profiles in saturation in matrix as a function of distance from the

fracture-matrix interface with different endpoint of krow at 30 days:.................................... 81


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different kro endpoints in matrix during 91 days: .................................................................. 82

Figure 6.19 Different relative permeability in matrix system by changing curvature of krow

....................................................................................................................................................... 83

Figure 6.20 Saturation profiles in saturation in matrix as a function of distance from the

fracture-matrix interface with different curvature of krow at 30 days: .................................. 84


different krow curvature in matrix during 91 days:.................................................................. 85

Figure 6.22 Schematic of calculation procedure for ∆krow:..................................................... 86

Figure 6.23 Saturation profiles in saturation as a function of distance from the fracture-

matrix interface with different capillary pressure in matrix at 30 days: .............................. 87



Figure 6.25 Δq as a function of time in models with different endpoint of krow .................... 89

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Nomenclature and Greek Symbols

𝐴𝑐𝑤 = cross-sectional area to flow defined as2ℎ𝑋𝑒, /m 2

𝐵𝑔 = formation volume factor of gas, rm 3/sm 3

𝐵𝑜 = formation volume factor of oil, rm 3/m 3

𝑐𝑡 = total compressibility, kPa −1

df = length of natural fracture, m

𝐸𝑔 = Gas expansion factor: 1/𝐵𝑔, sm 3/rm 3

f s = transfer function

ℎ = reservoir thickness, m

HF = hydraulic fracture

𝑘𝑚 = matrix permeability, mD

𝑘 𝐹 = hydraulic fracture permeability, mD

𝑘𝑓 = natural fracture permeability, mD

orowk = endpoint of oil relative permeability

𝑘 𝑟𝑜𝑔 = oil relative permeability

rwk = water relative permeability

𝐿𝐹 = hydraulic fracture spacing, m

𝐿𝑓 = natural fracture spacing, m

𝐿𝑤 = spacing between two horizontal wells, m

NF = natural (secondary) fracture

𝑛F = number of hydraulic fracture stages

𝑛f = number of natural fractures

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𝑃𝑏 = bubble-point pressure, kPa

𝑃𝑖 = initial reservoir pressure, kPa

𝑃𝑤𝑓 = minimum wellbore flowing pressure, kPa

Pcq = the difference in oil rate between models with different capillary pressure in matrix,

m 3/day

Dq = dimensionless wellbore rate

𝑅𝑠 = solution gas-oil ratio, m 3/m 3

s = laplace variable, t-1, s-1

𝑆𝑜𝑟𝑤 = residual oil saturation

𝑆𝑤 = water saturation

𝑆𝑤𝑖 = initial water saturation

𝑆𝑤𝑛 = normalized water saturation

SRV = stimulated reservoir volume

𝑤𝐹 = hydraulic fracture width, m

𝑤𝑓 = natural (secondary) fracture width, m

𝑋𝑒 = effective well length, m

Dey = dimensionless reservoir half-length

𝑥𝑓 = hydraulic fracture half-length, m

𝑦e = reservoir half-length, m

= interporosity transmissivity ratio

= storativity ratio, dimensionless

𝜇 = viscosity, cp

XV

𝜙𝑚 = matrix porosity

𝜙𝐹 = hydraulic fracture porosity

𝜙𝑓 = natural (secondary) fracture porosity

1

1 General Introduction

1.1. Overview and Background

Unconventional resources have become an increasingly important energy supply in North

America in the past two decades. Shale gas, tight oil, tight gas and coal-bed methane have been

widely explored and exploited as commercial resources around the world. The proliferation of

activity into shale plays has increased the production of shale gas. In North America, it makes

the contribution of shale gas to total natural gas production from less than 1% in 2000, to 39%

for the United States and 15% for Canada separately. (Stevens 2012; US Energy Information

Administration, 2013a).

Unconventional resources are in the unconventional reservoirs that are with low matrix

permeability in the order of nano darcies (10-6 mD) to micro darcies (10-3 mD) (Best and

Katsube, 1995), small porosity (less than 10%) and small pore size in the order of nano meters

(10-9 m) (Nelson, 2009). Large well-reservoir contact area is required to achieve commercial

production from the unconventional reservoirs (Ning et al., 1993).

The use of multilateral horizontal drilling in conjunction with multi-stage hydraulic fracturing

technologies has greatly expanded the ability to produce natural gas and oil profitably from

unconventional plays (Wang et al., 2008; Medeiros et al., 2010). The first application of

hydraulic fracturing can back to 19th century. In the 1970s, the developments of Devonian shale

in the eastern US foster the crucial technologies for unconventional resources, which including

horizontal drilling, multi-stage fracturing and slick water fracturing (US Energy Information

Administration, 2013b). After the 1990s, the advances in horizontal drilling and hydraulic

fracturing techniques, successful development of shale gas system such as Barnett Shale in

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Texas, US have led to increased exploitation and development internationally and large-scale

shale gas production in North America.

The decline in conventional resources and increasing demand for energy supply make oil

companies more active to seek out new unconventional reservoirs for commercial development.

However, development of unconventional formations still faces engineering challenges. Since

the hydraulic fracturing system creates complex fracture network along horizontal wellbore,

understanding the multiphase flow in the fracturing system is essential for reservoir estimation

and production forecast, which can eventually lead to better optimization of drilling and

production process. Moreover, because of the extremely small nano pore size in unconventional

reservoirs, capillary end effect will play a more important role in hydrocarbon recovery,

compared with inhigh permeability high porosity conventional reservoirs.

3

1.2. Problem Statement

Existing triple-porosity models typically assume sequential flow from matrix to micro fractures

and from micro fractures to hydraulic fracture. Modeling simultaneous depletion of a matrix

block into both micro and hydraulic fractures entails solution of a two-dimensional continuity

equation that is challenging by analytical or even semi-analytical methods. In addition,

production data analysis is an inverse problem with non-unique solutions. Analysis with

analytical models and type-curves provides deterministic and homogeneous estimates,

representing an average parameter value and rendering uncertainty analysis of fracture properties

difficult. In other word, it is hard to capture the uncertainties associated with reservoir

heterogeneity.

Field data usually show a rapid decline of oil production or low water flowback from fractured

horizontal wells. It has been hypothesized that multiphase effects such as phase blockage are

partly responsible for inefficient fracturing water and hydrocarbon recovery. Furthermore, recent

imbibition experiments indicate the significance of capillary pressure during multiphase flow in

unconventional rocks. In the late part of this thesis, we hypothesize that the discontinuity of

capillary pressure at fracture-matrix interface leads to significant phase blockage and relative

permeability effects, which in turn result in production decline. To test this hypothesis, we run a

series of simulation case studies to investigate the role of capillary discontinuity on multi-phase

production from fractured horizontal wells.

4

1.3. Research Objectives

In this thesis, we use a commercial reservoir simulator to compute the transient response in a

segment of a hydraulically fractured horizontal well with a triple-porosity model, where matrix

and fracture elements are explicitly discretized. Some limitations in analytical models such as

sequential flow, single-phase flow and fully-connected symmetric fractures are investigated

using the actual rate data from two tight oil wells completed in the Cardium and Bakken

formations. An early drop in production, as commonly observed in many tight oil wells, could be

attributed to gas dissolution with pressure decline and multiphase flow effects. Impacts of

pressure interference between natural fractures and inter-well fracture communication are also

investigated. Uncertainty in model parameter estimation is studied by assuming that results

obtained from analytical solutions would characterize the mean estimate of the corresponding

fracture parameters, additional heterogeneous models of natural/induced micro-fracture

properties including its total number and intensity (spacing) are assigned stochastically. These

models are subsequently subjected to flow simulations, and the variability in production

performance captures the sensitivity due to model parameter uncertainties.

In addition to this, the simulated saturation profiles confirm the existence of fracture-face effect,

which in principle is similar to the end-effect observed in laboratory coreflooding tests. Fracture-

face effect is caused by seeking capillary equilibrium in two systems with different capillary

pressures. The simulated production data indicate that fracture-face effect results in rapid

production decline. Less water blockage and higher production are observed in cases with lower

capillary contrast between fracture and matrix or higher relative permeability to the oil phase. It

is also observed that initial oil production increases slightly due to the displacement of oil by

water (wetting phase) near the fracture-matrix interface. This increase is more pronounced when

5

we increase the capillary pressure in matrix blocks. However, long-term production is hampered

as a result of increased water blockage.

6

1.4. Thesis Layout

Most of the research is using numerical reservoir simulation. Methodology consists of rate

transient analysis, history matching and sensitivity analysis.

The work in this thesis is divided into six chapters. Chapter 1 (the current chapter) provides the

background and the scope of this research including problem statement and some hypotheses.

Chapter 2 contains the introduction of modeling fractured well and transient flow. A detailed

literature review on multi-phase effects on production is also included in this chapter. An

overview of the geological features, reservoir heterogeneities and fracture characteristics for

typical tight oil reservoirs is included in Chapter 3. A discussion of the hydraulic fracturing

operation is also presented. This discussion serves as a basis for the numerical models employed

in this thesis. Chapter 4 presents the details of the proposed modeling methodology. Rate

transient analysis (RTA), Laplace transform and history matching are all explained in this

chapter. Chapter 5 comprises numerical investigation of limitations and assumptions of

analytical transient flow models. Several case studies are performed to highlight the non-

uniqueness of fracture characterization in production data analysis. Chapter 6 investigates two

multi-phase effects in tight oil production which are solution gas and capillary end effect. Some

sensitivity including relative permeability, bottom-hole pressure and capillary pressure are tested

in these two cases. Chapter 7 summarizes the major findings of the conducted research and

presents suggestions for future research on this topic.

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2 Literature Review

2.1 A Brief Background of Unconventional Reservoirs

Unconventional resources are hydrocarbon reservoirs which have low permeability and porosity,

include shale gas, shale oil, tight gas, tight oil and coalbed methane. Tight gas and oil are the

hydrocarbon gathered in small, poorly connected cavities between poorly porous sandstone.

Shale gas and oil are the hydrocarbon which still remain in the bedrock where it formed instead

of migrating to more permeable rock. Shale is characterized as a fine-grained, clastic

sedimentary rock composed of clay minerals and quartz grains (Crain, 2012). It has lower

permeability and porosity than tight sandstone reservoirs. The hydrocarbon in organic-rich shale

reservoirs usually trapped in the surfaces of the shale rock particles (Curtis, 2002). Until 2013,

there are 7,299 trillion cubic feet of shale gas and 345 billion barrels shale or tight oil which

located in 95 basins around the world are identified as technically recoverable resources around

the world (US Energy Information Administration, 2013b).

Stimulation techniques such as horizontal drilling, multi-stage fracturing and slick water

fracturing are required for achieving economic hydrocarbon production from unconventional

reservoirs. However, effective developments must consider the complex characteristics of

hydraulic fracturing system, which make the reserve estimation and production forecast more

difficult than conventional reservoirs. Accurate modeling and analysis on hydraulic stimulated

reservoirs are essential for better optimization of drilling and production process.

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2.2 Modeling and Analysis of Flow Response in Hydraulically-Stimulated Tight Reservoirs

Multi-fractured horizontal wells are required to enhance reservoir contact area in order to

achieve economic production rates (Ning et al., 1993). Modeling and analysis of flow response

in hydraulically-stimulated tight reservoirs are required for reserve estimation and production

forecast. Works including Warren and Root (1963), Kazemi (1969) and Swaan (1976) have

provided the basis for analyzing flow in a dual-porosity system. In these models, fluid flows

from the low-permeability matrix into a high-permeability fracture network, which is connected

to the production wellbore. El-Banbi (1998) extended the dual-media model for application in

stimulated tight reservoirs. Five distinct alternating linear and bilinear flow regimes representing

depletion in hydraulic fracture and surrounding matrix blocks can be identified (Bello, 2009).

Abdassah and Ershaghi (1986) developed a triple-porosity model for analysis of pressure

transient data, which could explain the anomalous slope changes observed during various

transition periods. Since most tight reservoirs are also naturally fractured, more comprehensive

triple-porosity models encompassing hydraulic fractures, natural fractures and matrix blocks

were developed recently (Abdullah A G and Iraj E, 1996; Liu et al., 2003; Wu et al., 2004;

Alahmadi, 2010, 2013; Dehghanpour and Shirdel, 2011; Abbasi et al., 2014; Ali A J et al., 2013).

In these models, natural or secondary fracture is considered as the third porosity system (Gale et

al., 2007), which can improve the stimulated reservoir volume as compared to a dual-porosity

system involving only hydraulic fracture and matrix (Rogers et al., 2010; Castillo et al., 2011).

The aforementioned analytical models have been employed extensively in the areas of pressure

transient (PTA) and rate transient (RTA) analysis. Many authors have also proposed simplified

semi-analytical analysis equations that can be used to estimate various system parameters such as

hydraulic fracture half-length and fracture-matrix contact area (Bello, 2009; Szymczak et al.,

9

2012; Ali et al., 2013). It should be emphasized that analytical or semi-analytical models are

essentially simplified solutions to the detailed governing equations. Unfortunately, production

data analysis is an inverse problem with non-unique solutions. Analysis with these analytical

models would typically yield a homogeneous deterministic estimate, representing an average

parameter value, but it fails to capture the uncertainties associated with reservoir heterogeneity

(Al-Ahmadi and Wattenbarger, 2011). At last, assumptions associated with most (semi-)

analytical models include single-phase flow (ignoring the effects of solution gas) and sequential

depletion from matrix to natural fractures and from natural fractures to hydraulic fracture (Al-

Ahmadi and Wattenbarger, 2011; Ali A J et al., 2013). These models also assume a fully-

connected natural fracture system, rendering detailed transient flow analysis in a triple-porosity

medium challenging.

In theory, numerical reservoir simulators can be used to estimate the transient flow response in a

triple porosity medium in which matrix blocks deplete into the two fracture networks

simultaneously within an arbitrary drainage volume; however, computational costs and

complexities of simulation models often hinder their efficiency in practice (Alkouh et al., 2012;

Kalantari -Dahaghi et al., 2012; Lee and Sidel, 2010). We demonstrate with production data

collected from two tight oil wells that the history matching process generates non-unique

solutions. The effects of boundary conditions, pressure interference and gas dissolution would

also introduce additional uncertainties in the parameters derived from analytical models. The

analysis workflow adopted in this study presents a practical framework for integrating numerical

simulations with analytical solutions to study the limitations of analytical transient flow models

and to quantify the uncertainty in production performance predictions.

10

2.3 Modeling and Analysis of Multiphase Flow Response in Tight Reservoirs

Unconventional reservoirs have received a significant attention recent years due to recent

advances in horizontal drilling hydraulic fracturing and the large amount of recoverable oil and

gas (Isaacs 2008). Multi-hydraulically fractured horizontal wells could enhance production by

connecting artificial fractures with micro fractures. Modeling and analysis of multiphase flow

response in tight reservoirs with multiple fracture systems are required for reserve estimation and

production forecast.

Works such as Clark (1962), Synder (1969) and Iwai (1976) have provided the basis for

analyzing flow in a multi-phase fractured system. In these models, a set of differential equations

which combine Darcy’s law and the law of conservation of mass for each phase describe

reservoir behavior. Based on this, Kazemi (1976) combined these flow equations to build a three

dimensional, numerical simulator to simulate the water and oil flow in fractured reservoirs.

Fetkovich (1980) developed type curve analysis method for oil wells based on the material

balance equations and rate-time equations (Fetkovich, 1973). Fraim and Wattenbarger (1988)

presented a method to analyze multiphase flow with Fetkovich (1980) type curves. Rossen and

Kumar (1992) considered the effect of aperture distribution and gravity, developed a two-phase

flow model using Effective Medium Approximation (EMA) (Kirkpatrick, 1973). Effect of

fracture relative permeabilities on natural fractured reservoirs is investigated using Rossen and

Kumar model. (Rossen and Kumar, 1994). Eker et al. (2014) modified a single-phase pressure

drop equation, and introduced a multiphase flow model for analyzing well performance of

fractured shale oil and gas reservoirs.

Brownscombe and Dyes (1952) performed water-oil imbibition experiments and observed oil

can be displacement by water imbibition. The experiment results coupled with estimates of the

11

extent of fracturing, concluded the capillary pressure could be a main recovery mechanism in the

highly fractured water-wet reservoir. Firoozabadi and Markeset (1992) performed drainage

experiment on stacked block and stacked slab rock samples and observed the capillary cross flow

enhanced the drainage rate. It is critical to know the capillary discontinuity in gravity drainage

(Horie et al., 1988; Labastie, 1990). Ignorance of fracture-face effect result in erroneous results

on relative permeability calculation in most of multiphase core flooding experiments (Qadeer et

al., 1991; Huang and Honarpour, 1998).). Rangel-German (2006) conducted water-air

imbibition and oil-water drainage experiments, concluded that capillary continuity play more

important role if the fractures are wider. Gupta et al. (2015) claimed that fracture-face effects

have minimal impact on field-scale recovery according to their corefloods results. In this paper,

we hypothesize that the discontinuity of capillary pressure at fracture-matrix may result in

production decline.

Although experimental studies are useful in understanding the system physics, they are also

costly and limited in study scale. Analytical models, though extensively employed in the areas of

pressure transient (PTA) and rate transient (RTA) analysis, are essentially simplified solutions to

the detailed governing equations; certain common assumptions include sequential depletion from

matrix to a fully-connected secondary fracture system and from secondary fractures to hydraulic

fracture (Al-Ahmadi and Wattenbarger, 2011; Ali et al., 2013) are often invoked. Numerical

reservoir simulators offer a viable alternative to estimate the transient flow response in a

complex medium in which fluids could flow into the two fracture networks simultaneously

within an arbitrary drainage volume.

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3 Geological and Production Characteristics of Tight Oil Reservoirs

In this chapter, geological characteristics, including those exhibited by complex multi-scale

heterogeneous fractured systems, of typical tight oil reservoirs are summarized. In addition,

completions/production technologies of horizontal wells and hydraulic fracturing are discussed,

as these techniques are commonly adopted to unconventional reservoir development. The

discussion would focus on two specific tight formations: Bakken and Cardium formations.

13

3.1 Geological Characteristics

Bakken Formation

Recently, the Bakken formation has been regarded as one of the largest contiguous deposits in

North America due to its huge oil accumulation (US EIA, 2014). It is an interbedded sequence of

rock unit from the Late Devonian to Early Mississippian age underlying large areas of

northwestern North Dakota, Southern Saskatchewan, northeastern Montana and southwestern

Manitoba (NoRDQuiST, 1953). The rock formation consists of three distinct members: lower

and upper shale and middle member which is shown in Fig 3.1.

Figure 3.1 Schematic north-south cross section showing the Bakken and adjacent formations (USGS, 2013)

The upper and lower shale are organic-rich marine shale and share similar characteristics. Due to

their low porosity and low permeability, upper and lower shale are acting as seals to the

generated hydrocarbon (Wiley et al., 2004). The middle member is the main productive target of

the current development. It is composed of marine sandstone or siltstone with large amount of

14

carbonate grains and cements (Cox et al., 2008).

Production in the Bakken can be divided into three development periods, that is, conventional

vertical drilling (1953-1987), horizontal drilling in the upper shale (1987-2000) and horizontal

drilling in the middle Bakken (2000-now) (LeFever, 2004).The application of horizontal well

and hydraulic fracturing treatments have caused a considerable increasing in Bakken production

since 2000 .

Although average porosity (5%) and permeability (0.04 md) are much lower than typical oil

reservoirs, the existence of vertical secondary fractures have greatly enhanced the efficiency of

horizontal drilling (Pitman et al., 2001). In this way, it is easy for a borehole to contact thousands

of meters of oil reservoir rock with only about 40 m thickness (Baars, 1972). Also, production is

increased by artificially fracturing the rock to have fracture with high conductivity for oil

flowing into well (Yedlin, 2008). It has been proven that horizontal well with hydraulic

fracturing is the most effective treatment to develop the middle Bakken formation (Tabatabaei et

al., 2009).

Cardium Formation

The Cardium Formation is a stratigraphic unit of Late Cretaceous age and is a major source of oil

and natural gas (Krause et al., 1994). It extends from northeastern British Columbia near Dawson

Creek, to western Alberta. Oil is produced from Cardium formation in central Alberta, while

natural gas is produced in western Alberta. The sandstones in formation have good storage

ability and thick overlying becomes stratigraphic traps, while black shale, the underlying, is good

source rock (Alberta Geological Survey, 2009).

Porosity of the Cardium formation is typically less than 9% with permeability being less than 1

md. Variation of porosity and permeability within the matrix can be ignored, considering their

15

intrinsically low values (Hoch et al., 2003). Horizontal well and hydraulic fracturing are also

employed to connect to secondary fractures and enhance reservoir contact area or stimulated

reservoir volume (Alberta Research Council et al., 1994).

16

3.2 Hydraulic Fracturing

Hydrocarbon in tight reservoir is difficult to be produced due to very low permeability in

formation (Norbeck, 2011). Many unconventional reservoirs have recovery rate of less than 2%

of OGIP which is much less compared with conventional reservoirs of 80% (King 2010; Moore

2010). Recently, hydraulic fracturing becomes a successful technology and widely used in tight

reservoirs to improve hydrocarbon production performance (Hossain and Rahman, 2008).

Hydraulic fracturing is a formation stimulation technique used to create additional permeability

by fracturing production formation with pressurized liquids (Veatch et al., 1989). Hydraulic

fracturing operations are always performed by several portions of the lateral in horizontal well

and fracturing of each portion is called a stage (DOE, 2009). Hydraulic fractures are induced in

two phases (Weijers, 1995). At first, preparations such as perforation on casing and creating

finger-like holes in formation are followed by pumping viscous fluids into well (Taleghani,

2009). A fracture is formed from perforation and develops into reservoir when bottom-hole

pressure is above breakdown pressure and fracture width becomes relatively large during

pumping (Cipolla et al. 2009). Next, fracturing slurry containing fluid and proppant is injected.

This slurry increases width and length of fracture and transports the proppant into the end of

fracture. Proppant is solid which can prevent the fractures from closing after finishing injection

(Gandossi, 2013). Proppant distribution, controlled by proppant selection, fluids viscosity and

fracture complexity, influences productivity of hydraulic fracture directly (Daneshy, 2004;

Cipolla et al. 2009). Fig 3.2 shows a comparison that a fracture with or without proppant after

pumping stops.

Cipolla et al. (2008) proposed that production enhancements in tight reservoir could be achieved

by complex fracture and average proppant concentration would decrease with fracture

17

complexity increasing. Cipolla et al. (2009) claimed that proppant transport and proppant-filled

height play important role in tight unconventional reservoirs. Using “waterfracs” which consists

of treated water and very low proppant concentrations in recent hydraulic fracturing treatments

has been successful in tight reservoirs (Fredd et al., 2000).

Figure 3.2 Proppant distribution in hydraulic fracture (Cipolla et al. 2009)

After slurry with proppant is pumped in, fluid with lower viscosity flows back out of the well, a

fracture with high conductivity is formed (Veatch et al., 1989). Typical widths of hydraulic

fractures are less than 0.6 cm, which is quite narrow. However, the effective length of hydraulic

fracture can goes to as long as 900 m (Buchsteiner et al., 1993).

Hydraulic fractures are tensile fractures, and they always open in the direction of least

resistance (Nolen-Hoeksema, 2013). Some geologic discontinuities (joints, bedding planes, faults

and stress contrasts) have significant influences on hydraulic fracture, including reducing total

18

length of hydraulic fracture by fluid leak-off or difficulty of proppant transport and placement

(Warpinski and Teufel, 1987). Micorseismic imaging of fracture is relied on detection of micro-

earthquakes or acoustic emissions associated with fracture (Urbancic et al., 1999). It can be used

to characterize the fracture network (including fracture spacing, conductivity, degree of

complexity, arimuth and fracture dimensions) and approximate fracture geometry (Fisher et al.,

2002; Mayerhofer et al., 2006; Nejadi et al., 2015). Fractures are responsible for the main part of

the permeability and mechanical closure of fracture is associated with pressure decline (Raaen et

al., 2001). In other word, pressure declined can be a result of reduced fracture size, i.e. the

pressure is less than or equal to the smallest principal stress in some part of the fracture (Fjar et

al., 2008). Decreasing the fluid pressure causes the fracture to close, and the permeability

decreases because of the smaller aperture (Walsh, 1981). Fracture closure can directly lead to

production decline. Hence, modelling of fracture geometry is important in hydraulic fracturing

for enhancing production in low-permeability reservoirs (Teufel and Clark, 1984).

For Bakken formation, horizontal well and hydraulic fracturing are used to achieve optimum

recovery (Miller et al., 2008). Phillips, et al., (2007) highlighted the importance of fractures with

high conductivity in Bakken horizontal wells and claimed that proppant characteristics are key

parameter for transverse fractures (Fig 3.3). Due to low permeability in Bakken, horizontal well

with multi-stage hydraulic fractures are required to increase oil production (Hassen et al., 2012).

Others have suggested that the length of a hydraulic fracture is the most important factor

influencing hydrocarbon productivity (Tabatabaei et al., 2009). It is also concluded that the

optimum fracturing job would be the fracturing treatment that will cross and connect as much of

the natural fracture system to the wellbore as possible. (Dahi, 2009).

19

Figure 3.3 Horizontal well with transversely intersecting fracture (Phillips, et al., 2007)

20

3.3 Natural Fracture (Secondary Fracture)

Secondary or natural fractures have been recognized as an important factor for hydrocarbon

recovery in unconventional reservoirs (Gale et al., 2014). Hydrocarbon recovery commonly

exceeds the rate which is expected from hydraulic-fracture stimulation of intact host rock alone

(Medeiros et al., 2010; Al-Ahmadi and Wattenbarger, 2011; Walton and McLennan, 2013; Yan

et al., 2013). Natural fracture can be generated by many different process include 1) local stress

perturbations; 2) regional burial; 3) stress change due to oil and gas generation or diagenetic

reactions; 4) tectonic palaeostress; 5) accommodation effects around major faults and folds; 6)

local structures and 6) stress release during uplift (Gale and Holder, 2010). The natural fractures

in unconventional reservoirs are poorly connected to each other, rendering the formations

economically unproductive. Without hydraulic fracturing stimulation, many tight gas reservoirs

have been estimated to recover less than 2% of original gas in place (OGIP) (King, 2010; Moore,

2010). The connectivity of natural fractures can be effectively reactivated by hydraulic fracturing

operations associated with complex microseismic event (Blanton, 1982; Warpinski and Teufel,

1987; Fisher et al., 2002, 2005; Dunphy and Campagna, 2011; Fisher and Warpinski, 2012),

which will ultimate increase the estimated ultimate recovery (EUR) of tight gas reservoir to

nearly 50% (King, 2010). Lack of site-specific natural fracture information is an impediment to

understanding microseismic patterns and evaluating the hydraulic fracturing efficiency.

The overall productivity of a well and the role of natural fractures play as mechanical

discontinuities and flow conduits depend on the number of fractures, fracture connectivity, and

fracture characteristics (Wei and Economides, 2005). Understanding the characterization of the

natural fracture has been considered as an important factor for determining the hydraulic

fracturing, designing the well completion strategies and estimating the hydrocarbon production

21

performance (Aguilera, 2008). Fracturing properties such as fracture length, fracture width,

fracture permeability, fracture conductivity, spatial arrangement, strength and cohesion and

fracture mode (opening or shear) can be inferred from a variety of data including production

data, core samples, resistivity image logs and cuttings and mud log data from drilling process

(Norbeck, 2011).

Fracture abundance can be evaluated by fracture intensity or spacing, which is defined as number

of fractures of a given size per unit of rock (scanline length, outcrop area or rock volume)

(Ortega et al., 2006). It is more accurate when more fractures are sampled in a core or outcrop.

However, sparse fractures are difficult to be captured in a limited area of outcrops, or the fracture

clustering is easily misleading the fracture abundance observations. Another method to measure

the fracture abundance is spacing, which is defined as separation between adjacent fractures,

calculated as the inverse of intensity (Narr, 1996). Spacing is effectively analyzed in horizontal

well data sets collected from a large outcrop. Fracture density or P32 is another measure of the

fracture abundance, which is defined as the ratio of fracture surface area to rock volume

(Dershowitz and Einstein, 1988). Fracture trace data in borehole images or whole core can be

used to provide sufficient data sets for quantification of fracture density (Barthélémy et al.,

2009). However, it is difficult to calculate or measure the fracture surface area and the

micrometer wide fractures are difficult to be identified on chattered surfaces of the core. Vertical

intensity of natural fractures are varies from 7 to 160 fractures per 100 ft for the unconventional

reservoirs (Gale et al., 2014).

Geometry of an individual fracture is defined by its size (length and aperture) and orientation

(i.e., dip and azimuth angle). Many unconventional formations exhibit a wide range of fracture

size. Kinematic apertures range from micrometer to millimeter for both tight and shale reservoirs

22

(Laubach and Gale, 2006; Gale et al., 2014). The population of natural fracture usually follows a

power-law size distribution (Marrett et al., 1999; Ortega and Marrett, 2000). Fracture length can

be observed on some outcrop or quarry, but it is limited by the fracture exposure and the fracture

may be not well preserved because of weathering (Olson, 2003). The maximum length observed

by Gale et al.(2014) is around 40 m length with about 3 m height and 0.5 to 1 mm thick in

Marcellus quarry outcrop. Natural fractures in shale are typically parallel arranged to the bedding

plane, and propagate along the plane perpendicular to the least-compressive principal stress

(Secor, 1966). Similar fracture orientations may indicate they formed under a similar stress

regime (Hodgson, 1961; Hancock, 1985). Systematic regional patterns (consistent or gradually

varying fracture patterns) of fracture strike has been observed in many basins include the

Appalachian Plateau (Nickelsen and Hough, 1967; Engelder et al., 2009), the midcontinent New

Albany (Ault, 1989) and the Michigan Basin Antrim Shales (Apotria et al., 1994). However,

some formation also contain unsymmetrical fracture strike patterns. For example, fracture sets

are not always follow the same sets in the outcrop in parts of the New Albany Shale (Gale and

Laubach, 2009). This can be explained by weathering changes (Fidler, 2011), stress-field rotation

(Rijken and Cooke, 2001).

Multiple orientation fracture patterns can be observed in shales with complex regional loading

patterns and isotropic stress (Tuckwell et al., 2003; Fidler, 2011). Formation with such highly

interconnected fracture pattern shows better production performance compare to the similar

formation with a similar number of fractures but not well-connected (Philip et al., 2002).

Natural fracture permeability of a tight reservoir can be varied from 3 to 100 mD (Rubin, 2010;

Cheng, 2012; Fakcharoenphol et al., 2013; Wattenbarger and Alkouh, 2013). Natural fracture

porosity can be typically varied from 0.1% to 8% in unconventional formations (Nelson, 1985;

23

Weber and Bakker, 1981). The natural fracture porosity can be underestimated because the size-

dependent sealing patterns make the larger fractures tends to host great porosity (Laubach,

2003). Fracture porosity is far more compressible than normal matrix porosity (Ostensen, 1983).

Fracture porosity usually decreases linearly with the log of confining stress (Walsh, 1981; Walsh

and Grosenbaugh, 1979). Laboratory studies using tight sandstones shows the linear relation of

fracture porosity and the log of confining stress (Jones and Owens, 1980; Sampath, 1982). The

permeability of jointed natural fractures can be significantly reduced due to the stress decreasing.

The fracture wound not be fully closure and the permeability is still higher than matrix

permeability (Gutierrez et al., 2000; Cho et al., 2013). For partly cemented natural fractures, the

fracture is naturally propped with cement, which can prohibit closure from the stress decrease

during production.

Not all natural fractures contribute to production. It depends on whether they are connected to

the hydraulically fracture or activated according to the stress conditions (Cipolla et al., 2009).

Natural fractures with high permeability and well-connected to the hydraulic fractures, usually

have a positive effect on hydrocarbon recovery (Curtis, 2002; Engelder et al., 2009). High

permeability natural fractures can also be a hindrance to hydrocarbon recovery (Dyke et al.,

1995). For a conductive fracture system, water may only drain high permeability fractures but

not low-permeability oil-saturated pores. Conductive fractures may also result in early

breakthrough of water and hydrocarbon. For long-term production, natural fractures might close

or be less contact with the hydraulic fracture system due to the fluid pressure depletion (Pagels et

al., 2012; McClure, 2014). The horizontal natural fractures may act as barriers to hinder the

hydraulic fracture propagate vertically when they intersect (Weng et al., 2011).

It is hard to measure natural fractures properties (such as permeability and length) by using log

24

imaging. In tight reservoir it is not practical to use well test analysis to estimate natural fracture

properties because low reservoir permeability slows down the reservoir responses and testing

time is long (Nejadi et al., 2014). Alternatively, analysis of dynamic production data has been

adopted for estimations of fracture properties. However, production data analysis is an inverse

problem with non-unique solutions (Nejadi et al., 2014). In this thesis, we have an integration of

analytical solution and numerical solution to quantify uncertainties in reservoir and fracture

properties.

25

4 Methodology

A commercial black-oil simulator (Computer Modeling Group, 2013) is used to construct a 2-D

numerical model composed of three interacting media: natural fractures (NF), matrix blocks and

hydraulic fractures (HF). There are no restrictions regarding the flow direction and the number

of fluid phases. If we assume that hydraulic fracture stages are evenly spaced and symmetric, the

simulation domain can be represented simply by a segment of a horizontal well with two bi-wing

hydraulic fractures as shown in Fig 3.1. The top view of a horizontal well oriented along the x-

direction with two additional fracture systems is illustrated. Perforation is placed in the center of

the simulation model where the hydraulic fracture is intersecting with the horizontal well. Some

natural fractures are positioned evenly at each side of horizontal well.

Figure 4.1 Top view of a horizontal well in triple-porosity model

The model depicts essentially a “single-porosity” medium, where matrix and fracture systems are

explicitly discretized, and distinct properties are assigned to grid blocks that belong to each of

the three systems. Due to the vast difference in the width dimensions of fractures and matrix, a

logarithmic local grid refinement (LGR) scheme is employed to discretize regions around the

Hydraulic FractureNatural Fracture

Well

26

fractures, enhancing stability of the numerical solution while accurately capturing the transient

responses within the fractures. The grid size varies from 0.05 m in the fracture cells to 10 m in

the matrix cells that are located far away from the fracture.

4.1 Rate Transient Analysis

Parameters in the simulation model should be assigned according to known field

observations/operating variables, as well as results obtained from RTA analysis. Analytical

models such as Bello (2009) or Ezulike et al. (2015) can be used to estimate the system

parameters pertinent to a dual-porosity or triple-porosity system, respectively. After

identification of the observable flow regimes from the data, analysis equations can be applied to

compute properties such as fracture permeability, fracture intensity, reservoir half-length (ye) or

stimulated reservoir volume (SRV). It is likely that a number of combinations of different

parameter values could result in the same production data match.

27

4.2 Analytical Solution: Using Laplace Transform

The assumption of evenly-distributed hydrauclic fracturing stages is explored in this section.

Analytical models are applied to the Cardium well data to estimate a number of unknown

parameters. First, the dual-porosity model of Bello (2009) is used to estimate the unknown

hydraulic fracture properties such as half-length and permeability of hydraulic fracture. The rate-

normalized pressure or RNP plots of three dominant flow regions are shown in Fig 3.2. For the

bilinear region, a plot of RNP against √𝑡4

(Fig 3.2a) yields a straight-line slope 𝑚1, which is

substituted into Bello’s region 2 analysis equation to obtain kF:

√𝑘𝐹 = 966 ×𝜇𝐵𝑜

𝐴𝑐𝑤√𝐿𝐹 √

1

𝑘𝑚(∅𝑐𝑡)𝑡𝜇

4×

1

𝑚1 (1)

Where kF = hydraulic fracture permeability; LF = spacing between fracture stages; = fluid

viscosity; ct = total compressibility; = porosity; Bo = oil formation volume factor; km = matrix

permeability; and 𝐴𝐶𝑊 = cross-sectional area to flow defined as 2ℎ𝑋𝑒 (Xe = effective well

length). The half-length of hydraulic fracture (xF) is estimated by substituting the straight-line

slope 𝑚2obtained from a plot of RNP against √𝑡 (Fig 3.2b) for the linear flow regime into

Bello’s region 4 analysis equation:

𝑥𝐹 =5.8𝐵𝑜𝐿𝐹

𝐴𝑐𝑤√

𝜇

𝑘𝑚(∅𝑐𝑡)𝑡×

1

𝑚2 (2)

The values of kF and xF are estimated to be 1090 mD and 112 m, respectively.

Next, data from the boundary-dominated (linear pseudo steady state) flow regime is analyzed

with the model described by Siddiqui et al. (2012). Assuming hydraulic fractures are fully

28

connected between wells (i.e., ye = xF), ye and km can be estimated. A plot of RNP against

material balance time (MBT) (Fig 3.2c) would yield a straight-line slope 𝑚𝑠𝑠 and intercept 𝑏𝑠𝑠:

𝑥𝐹 =𝐵𝑜

2(∅𝑐𝑡)𝑚ℎ𝑛𝐹 𝐿𝐹×

1

𝑚𝑠𝑠 (3)

𝑘𝑚 =𝜇𝐵𝑜𝐿𝐹

24𝑛𝐹 𝑥𝐹ℎ×

1

𝑏𝑠𝑠 (4)

Where h = formation thickness; xF and km are evaluated to be 101 m and 0.034 mD, respectively.

It is noticed that the correlation between RNP and MTB is not very strong in Fig 3.2c. This

might introduce uncertainties in the estimation of xF and km using this method. It is expected that

some discrepancies would exist between xF values estimated using the two different analytical

models. Unlike Siddiqui’s model, estimation of xF form Bellos’s model depends on the value of

km. In this analysis, a permeability value of 1.34 mD was measured from core samples, and it

was used as km in Bellos’s analysis. This km value, however, is substantially larger than the

estimate from Siddiqui’ model, since micro fractures might exist in the core samples and true km

is difficult to measure. Therefore, the value of km obtained from Siddiqui’s model could serve as

another estimate of the true matrix permeability.

Finally, a quadrilinear flow model (QFM) proposed by Ezulike and Dehghanpour (2013) is

applied to estimate additional unknown parameters including number of natural fractures (nf),

natural fracture permeability (kf), distance between two natural fractures (Lf). Laplace transform

was used to solve a set of governing equations that describe simultaneous depletion of single-

phase flow from matrix to natural fracture and hydraulic fracture under constant bottom-hole

pressure or constant rate at the inner boundary. Type-curve solutions are subsequently

constructed by inverting the solutions from the Laplace space (s) to real time numerically

(Ezulike and Dehghanpour, 2013). Eqs.5-14 are the solutions in Laplace space for the

29

dimensionless rate at the wellbore with constant bottom-hole pressure.

De

Dyssfs

ssfq

coth (5)

Where

sfsfs

ssfssfs

sf mmFmAC

ff

FfAC

F tanh3

tanh3

,, (6)

And

FmAC

mm

ssf

,

23

(7)

And

fmAC

m

fmAC

m

FfAC

fmAC

FfAC

f

f

ss

ssf

,

1

,

1

,

,

,

3tanh

33

(8)

Where

cw

eDe

A

yy (9)

tt

FtF

c

c

(10)

tt

mtm

c

c

(11)

cwF

m

f

fmAC Ak

k

L

1

2,

12 (12)

cwF

m

F

FmAC Ak

k

L

2

2,

12 (13)

cwF

f

F

FfAC Ak

k

L2,

12 (14)

Here, 1 and 1 are weighting parameters, which control the fraction of fluid in the matrix that

depletes into natural fracture and hydraulic fracture, respectively, during production; km1 and km2

denote matrix permeability in flow to natural fracture and hydraulic fracture, respectively. A

match between QFM type-curves and production data is illustrated in Fig 3.2d.

30

Figure 4.2 Analysis of the Cardium well production data with analytical models

(a) –RNP against √𝒕𝟒

for bilinear flow; (b) – RNP against √𝒕 for linear flow;

(c) – RNP against MBT for boundary-dominated flow; (d) – QFM type-curve matching

The same analysis is repeated for the second well completed in the Bakken formation. Results

for xF are in good agreement with those reported in the literature. Duhault (2012) and Quirk et al.

(2012) have observed 60 m < xF < 300 m in Cardium formation, while O’Brien et al. (2012)

estimated 135 m < xF < 275 m from micro-seismic studies in the Bakken formation.

(a) (b)

0.E+00

5.E+02

1.E+03

2.E+03

2.E+03

3.E+03

3.E+03

4.E+03

4.E+03

0.E+00 1.E+03 2.E+03 3.E+03 4.E+03 5.E+03 6.E+03 7.E+03

RN

P (

kPa/

(sm

3/d

ay))

t (hr)

0.E+00

5.E+00

1.E+01

2.E+01

2.E+01

3.E+01

3.E+01

0.E+00 2.E-01 4.E-01 6.E-01 8.E-01 1.E+00

1/q

D

tD

(c) (d)

31

4.3 History Matching and Sensitivity Analysis of Simulation Models

Additional adjustment or tuning of estimates derived from analytical models is required to

achieve a final production history match with the simulation models. It is assumed that the

results obtained from this history-matching process would characterize primarily the mean

values of the corresponding fracture parameter distributions. A series of stochastic models are

subsequently constructed and subjected to flow simulations in order to assess the uncertainties in

fracture intensity (spacing) and the impacts of different assumptions associated with the

analytical models. The variability (spread) in the simulation predictions would capture the

sensitivity due to uncertainty in these model parameters.

32

5 Numerical Investigation of Limitations and Assumptions of Analytical

Transient Flow Models

The RTA and history matching procedures described in the previous section have been applied in

two case studies, where production data from two horizontal wells completed in the Cardium

formation and Bakken formation are examined. Known model parameters summarized from field

reports are shown in Table 5.1 (Ezulike and Dehghanpour, 2013). A few other unknown

parameters are assigned based on typical values for these formations (Hlidek and Rieb, 2011;

Alcoser et al., 2012; Clarkson and Pederson, 2011) and RTA, as shown in Table 5.2. The value

of fracture conductivity (kF × wF) used in this study is comparable to typical field observation

(Cinco, 1978). Although the assumed value for fracture porosity (F) appears to be high in

comparison to field observation, a sensitivity analysis reveals that the effects of F on the

numerical solution are minimal. This is because porosity influences primarily the accumulation

term in the governing equations, and its impacts are subdued in a slightly-compressible system

(e.g., oil). On the other hand, fracture permeability, kF, has a strong influence on the simulated

flow response, as it is used in the flux calculations. Table 5.3 is range of parameters estimated

from RTA. Several cases for assessing the uncertainties and limitations of analytical models are

presented next.

Table 5.1 Known Field Data

Parameter Symbol Cardium Well Bakken Well Unit

Formation volume factor of oil

𝐵𝑜 1.221 1.329 rm 3/m 3

Viscosity 𝜇 1.13 0.5643 cp Initial reservoir pressure 𝑃𝑖 15575 46884 kPa

Minimum wellbore flowing pressure

𝑃𝑤𝑓 1000 1000 kPa

Total compressibility 𝑐𝑡 1.54 × 10−4 2.51 × 10−6 kPa −1

Matrix permeability 𝑘𝑚 - 0.0005 mD

Matrix porosity 𝜙𝑚 0.12 0.09 -

33

Effective well length 𝑋𝑒 1370 1707 m

Reservoir thickness ℎ 7 5.8 m Number of hydraulic

fracture stages 𝑛F - 16 -

Table 5.2 Unknown Data – Assumed based on typical values observed in Bakken reservoirs

Parameter Symbol Value Unit

Natural fracture permeability

𝑘𝑓 500 mD

Hydraulic fracture permeability

𝑘 𝐹 1000 mD

Natural fracture porosity 𝜙𝑓 0.6 -

Hydraulic fracture porosity

𝜙𝐹 0.8 -

Water saturation 𝑆𝑤 0.2 -

Hydraulic fracture width 𝑤𝐹 0.01 m Natural fracture width 𝑤𝑓 0.00005 m

Hydraulic fracture spacing

𝐿𝐹 106 m

Hydraulic fracture half-length

Natural fracture length

𝑦𝑒 𝑑𝑓

200 106

m m

Table 5.3 Unknown Data – Estimated based on rate transient analysis (RTA)

Parameter Symbol Value Unit

Natural fracture permeability

𝑘𝑓 104 − 520 mD

Hydraulic fracture permeability

𝑘 𝐹 245 − 1224 mD

Number of natural fracture

𝑛𝑓 6-19 -

Hydraulic fracture half-length

𝑦𝑒 141-235 m

34

5.1 Configuration of Multistage Hydrualic Fracturing

Next, a series of dual-porosity simulation models with four stages of hydraulic fracture are

constructed to investigate the uncertainty in spacing between fracture stages on production

performance of the Cardium well. Four different configurations or sets of LF are tested (Table

5.4), and the corresponding pressure distributions at the end of production history (250 days) are

illustrated in Fig 5.1. Fig 5.2 compares the production performances of all four cases, which

deviate from each other with time. As expected, oil rate for the Base Case (where fracture stages

are uniformly spaced) is the highest because pressure depletion and drainage is the most

effective. In fact, a difference of 30% is observed between the Base Case and Case 0-3. This

observation implies that if an analytical model assuming symmetry and uniform spacing between

fracture stages is used to analyze production data from a well with asymmetric fracture stages

and uneven spacing, a significant underestimation of the fracture half-length (and SRV) would be

expected. The analytical solution, though not realistic, has provided an alternative (non-unique)

estimation, which could be sufficiently useful for production forecast; however, underestimation

of fracture half-length or SRV could potentially impact future field development decisions such

as placement of nearby wells to maximize drainage.

Table 5.4 Spacing between the four stages of hydraulic fractures for the Cardium well in Case A

Case Number Distribution of LF Unit

Case 0-1 146.4, 2.4, 147.6 m

Case 0-2 20.4, 146.4, 2.4 m

Case 0-3 87.6, 79.2, 69.6 m

Base Case 71.7, 71.7, 71.7 m

35

Figure 5.1 Pressure (in kPa) distribution at the end of production history (250 days) for the Cardium well with

different distribution of LF

(a) Case 0-1; (b) Case 0-2; (c) Case 0-3; (d) Base Case.

P

0 100 200 300

0 100 200 300

-20

0-1

00

0

-20

0-1

00

0

0.00 145.00 290.00 feet

0.00 45.00 90.00 meters

File: case a-1.irfUser: Min YueDate: 2014/6/4

Scale: 1:2213Y/X: 1.00:1Axis Units: m

2,178

3,520

4,862

6,204

7,546

8,889

10,231

11,573

12,915

14,257

15,600

Pressure (kPa) 2007-05-10 K layer: 1

P

0 100 200 300

0 100 200 300

-20

0-1

00

0

-20

0-1

00

0

0.00 145.00 290.00 feet

0.00 45.00 90.00 meters



2,177

3,520

4,862

6,204

7,546

8,889

10,231

11,573

12,915

14,257

15,600


P

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0 100 200 300

-20

0-1

00

0

-20

0-1

00

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2,139

3,485

4,831

6,177

7,523

8,869

10,215

11,561

12,908

14,254

15,600


(a) (b)

(c) (d)

36

Figure 5.2 Log-log plot of daily oil rate as a function of time for the Cardium well in Case A over 250 days

1

10

100

1 10 100 1000

Oil

Rat

e (

m3

/day

)

Time (day)

Base Case Case 0-1 Case 0-2 Case 0-3

Difference of 30%

37

5.2 Intensity/Spacing of SecondaryFractures

Approximate estimates of hydraulic fracture half-length (xf) and spacing between natural

fractures (or the number of natural fractures, nf) are obtained from production data analysis with

the triple-porosity analytical model described by Ezulike and Dehghanpour (2015). As discussed

in the previous sections, solutions to production data analysis are non-unique, and many different

combinations of these two parameters could yield the same data match. Therefore, based on the

ranges of values derived from RTA, a set of simulation cases with different number of natural

fractures (up to 20) are built and later matched with the production data. When the number of

natural fractures becomes zero, the model converges to a dual-porosity model. To match the

production data, different combinations of xf and nf are obtained. The final results for the

Bakken well are tabulated in Table 5.5 and presented graphically in Fig 5.3.

Table 5.5 History-matched hydraulic fracture half-length and the corresponding number of natural fractures for

the Bakken well

Dual-Porosity Case

Case 1 Case 2 Case 3 Case 4 Case 5

nf 0 2 4 8 12 20

xf 315 280 250 200 150 114

38

Figure 5.3 Relationship between xf –nf for the Bakken well

Interestingly, as the number of natural fractures increases, the history-matched hydraulic fracture

half-length is reduced. This is because natural fractures provide additional interface for matrix

depletion. In one extreme scenario where natural fractures are absent (representing a dual-

porosity system), a maximum hydraulic fracture half-length of 315m is required to achieve a

reasonable history match. This is almost three times the size in comparison to the other extreme

scenario where there are 20 natural fractures and a hydraulic fracture half-length of 114m. This

observation not only highlights the non-unique nature of production data analysis, but it also

demonstrates the important role of nf for production enhancement in a triple-porosity system.

In order to carry out further sensitivity analysis of other model parameters, a base case with nf =

8 and xf = 200 m is selected. A schematic of this base case is shown in Fig 5.4. Fig 5.5 shows the

pressure distribution for the base case at the end of production history after 4.5 years. Fig 5.6 is a

log-log plot of producing oil rate versus time for this base case along with the actual field data.

The overall trend of the production decline of the base case is in good agreement with the actual

0

50

100

150

200

250

300

350

0 5 10 15 20 25

x f(m

)

nf

39

field observations, but it fails to accurately predict two spikes in the oil production. The first

peak at 17 days corresponds to a drop in the bottom-hole pressure, while the second peak at 172

days is the result of reopening the well after a temporary shut-in. In this model, the bottom-hole

pressure is assumed to be constant.

40

Figure 5.4 Schematic of the base case with eight natural fractures for the Bakken well

Figure 5.5 Pressure (in kPa) distribution for the base case (Bakken well) at the end of production history (4.5

years)

Natural Fracture

Hydraulic Fracture

Well

1

2𝐿𝐹

𝑑𝑓

𝐿𝑓

𝑥𝑓𝑦𝑒

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8 Natural Fractures

Hydraulic Fracture

Well

Perforation

41

Figure 5.6 Log-log plot of daily oil rate as a function of time for the Bakken well

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Filed Data Base Case

42

5.3 Non-Sequential Flow

In all analytical models, a series of 1-D sequential flows are assumed: fluid flows from matrix to

natural fractures then from natural fractures to hydraulic fracture. Several simulation models are

built to test this assumption for the Bakken well. All parameters are the same as described at

beginning of this chapter; however, the i-direction permeability in matrix is reduced to nearly

zero to simulate the sequential flow conditions. Therefore, pressure drop between natural fracture

and matrix is uniform along the entire natural fracture. Fig 5.7 shows the pressure distribution for

the sequential case at the end of production history after 4.5 years. The drainage area in matrix is

reduced substantially, as compared with the base case shown in Fig 5.5, due to zero flux between

matrix and hydraulic fracture.

Figure 5.7 Pressure (in kPa) distribution for the sequential case (Bakken well) at the end of production history

(4.5 years)

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42,378

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43

Fig 5.8 compares the oil rate versus time for the sequential case and the base case. It can be

observed that the sequential model underestimates the oil rate as compared with the non-

sequential model, especially at early time. This observation implies that history matching with

analytical models would overestimate nf and xf to compensate the ignorance of matrix-hydraulic

fracture communication.

Figure 5.8 Log-log plot of daily oil rate as a function of time for the base case and the sequential case (Bakken

well)

To verify this hypothesis, more natural fractures are added to this sequential-flow case; two

additional models with 12 and 20 natural fractures are built. Fig 5.9 compares the daily oil rate

versus time for the base case and all three sequential cases (nf = 8, 12 and 20). As the number of

natural fractures increases the response of sequential flow case converges to that of the

simultaneous flow case (base case) with only 8 natural fractures. However, the difference

between the three sequential flow models with different values of nf is not obvious at early time

scales. This is because during early time, flow from hydraulic fracture to well contributes

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Field Data Base Case Base Case(low kx)

Difference between non-

sequential flow and

sequential flow model

44

primarily to the production. Therefore, increasing the number of natural fractures does not have a

significant influence on early time production. However, as production continues, drainage from

natural fractures and matrix becomes important; increasing the number of natural fractures in the

sequential model would enhance the contact area with the matrix and allow increased drainage

from matrix to natural fractures. This would essentially compensate for the reduction in fluid

transfer from the matrix to hydraulic fracture due to the near-zero permeability in i-direction.

Figure 5.9 Log-log plot of daily oil rate as a function of time for the base case and all sequential cases (Bakken

well)

Fig 5.10 shows the daily oil rate versus time for the non-sequential and sequential cases with

certain number of natural fractures (e.g., nf = 8 and 20). A transient flow in the hydraulic fracture

system is observed at early time. This is followed immediately by a bilinear flow (1/4 slope)

period in hydraulic fracture and natural fracture. Another short transient linear flow emerges,

which is followed by a period of bilinear flow (1/4 slope) in matrix and natural fracture. This

sequence of observable flow regimes is similar to the model #3 described by Al-Ahmadi (2011).

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Base Case (8NFs) Sequential Flow (8NFs) Sequential Flow (12NFs) Sequential Flow (20NFs)

Increasing number of

natural fractures

45

It is obvious that the difference between non-sequential and sequential with twenty natural

fractures is much less than the difference between two cases with eight natural fractures. In other

word, the difference between the two models diminishes as nf increases. That means the impact

of sequential flow assumption is less important when nf increases. This result suggests that for

forecasting gas or oil production production, especially at late time scales, sequential triple

porosity models may work for situations when nf is high enough. This observation also suggests

that analytical solutions may lead to an overestimation of nf or conductivity of the secondary

fracture network.

(a)

(b)

Figure 5.10 Log-log plot of daily oil rate as a function of time for non-sequential and sequential cases

(Bakken well): (a) –nf = 8; (b) – nf = 20.

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Base Case (8NFs) Sequential Flow (8NFs)

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Base Case (20NFs) Sequential Flow (20NFs)

46

5.4 Non-Symmetric Drainage

In most analytical models, the spacing between natural fractures and length of natural fractures

are assumed to be uniform. Hence, for nf = 8 in the history-matched base case, a value of natural

fracture spacing (Lf) of 44.4 m should be considered as an average or mean estimate, while the

length of each natural fracture is approximately 106 m (spacing between individual hydraulic

fracturing stages).

Firstly, a series of models (Case A) for the Bakken well with varying spacing between the eight

individual natural fractures are employed to investigate the impact of local heterogeneity of Lf on

production prediction. Four different distributions of Lf are tested, and their values are

summarized in Table 5.6. The degree of variability (heterogeneity) increases from Case A-1

through Case A-4.

Table 5.6 Spacing between natural fractures for the Bakken well in Case A

Case Number Distribution of Lf Unit

Case A-1 22.2 13.6 13.6 133 35.2 133 13.6 13.6 22.2

m

Case A-2 28.4 28.4 28.4 28.4 28.4 28.4 97.6 66 68

m

Case A-3 22.2 22.2 22.2 22.2 22.2 22.2 22.2 22.2 222.4

m

Case A-4 4 4 4 4 4 4 4 4 340 m

In Case A-1 (Fig 5.11a), four natural fractures are placed on each side of the horizontal well,

with one being very close to the perforation point while the other three fractures are located

much further away. The pressure map at the end of 4.5 years and the corresponding histogram of

Lf for Case A-1 are shown in Fig 5.11a. Results for Case A-2, which consists of six evenly-

47

distributed natural fractures on one side and two more on the other side, are shown in Fig 5.11b.

Fig 5.11c shows a case where all eight natural fractures are located on the same side of the

horizontal well. Fig 5.11d illustrates an extreme scenario where all natural fractures are located

on the same side of and far away from the well. It is clear that larger pressure drop, and thus

more drainage, can be experienced in the areas close to the natural fractures.

Figure 5.11 Pressure (in kPa) distribution of the Bakken well at 4.5 years and the histogram of Lf for

(a) – Case A-1; (b) – Case A-2; (c) – Case A-3; (d) – Case A-4.

Fig 5.12 compares the match with production history for the four cases, and the differences

between each case at early times shown in Fig 5.12a are not noticeable. However, when

P

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1,749

6,266

10,783

15,299

19,816

24,333

28,849

33,366

37,883

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46,916


0

1

2

3

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5

Fre

qu

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cy

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1,766

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33,371

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01234567

Fre

qu

en

cy

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1,755

6,271

10,788

15,304

19,820

24,336

28,852

33,368

37,884

42,400

46,916


0

2

4

6

8

10

Fre

qu

en

cy

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0

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1,707

6,228

10,749

15,270

19,791

24,312

28,833

33,353

37,874

42,395

46,916


0

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4

6

8

10

Fre

qu

en

cy

Spacing (m)

(a) (b)

(c) (d)

48

comparing the production performance from 1000 days to 20 years illustrated in Fig 5.12b, the

effect of natural fractures becomes dominating. It is obvious that the differences between these

five profiles become more pronounced with time. It can be noted that oil rate for Case A-4 is

much lower than that for other cases because pressure depletion and drainage is much less

effective on the side of the horizontal well where the natural fractures are absent. The base case,

on the other hand, gives the highest production rate at all times because natural fractures are

evenly distributed, allowing more effective drainage across the entire domain.

(a)

(b)


0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Base Case Case A-1 Case A-2 Case A-3 Case A-4

0.1

1

1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)


49

(a) – entire history for 20 years; (b) – late-time history after 1000 days.

Next, the four cases are repeated with matrix permeability increased by approximately 5 times

(i.e., km = 0.0026 mD) to investigate the interplay between matrix permeability and heterogeneity

in natural fracture spacing. Fig 5.13 shows the production history for these four new cases, and

the differences among them are negligible.

(a)

(b)

Figure 5.13 Log-log plot of daily oil rate as a function of time of the Bakken well for Case A with higher matrix

permeability (0.0026mD)

(a) – entire history for 20 years; (b) – late-time history after 1000 days.

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)


0.1

1

1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)


50

One possible explanation is that for this particularly high matrix permeability, specific locations

of a fixed number of natural fractures do not have significant impacts. With high km, relatively

uniform pressure depletion and oil drainage can be achieved readily, irrespective of the NF

locations. In other words, the contribution of natural fractures is not essential. On the other hand,

when km is low, drainage in regions with low NF density remains limited, significantly

hampering the ensuing production. Overall, the comparative analysis suggests that the impact of

natural fracture spatial distribution is more pronounced for low-permeability reservoirs.

As discussed before, most analytical models assume that all natural fractures are uniform in

length. In the next set of models (case B), length of individual natural fractures is varied,

allowing only a portion of the natural fractures to be fully connected. The case with natural

fractures length being half the distance between two hydraulic fractures (Rl =0.5) is chosen as the

base scenario here. Rl. is defined as the ratio of natural fracture length df to hydraulic fracture

spacing LF. A series of models (Cases B-1 to B-3) are constructed where the length of individual

natural fractures is sampled from different distributions with the same average (mean) value of

50 m. The idea is to ensure that average length of the eight natural fractures remains identical.

Three different distributions, as summarized in Table 5.7, are tested to investigate the effects of

local heterogeneity in natural fracture length on production performance. All cases have the same

average natural fracture length as in the new base case.

Table 5.7 Natural fracture length for the Bakken well in Case B

Case Number New Base Case

Case B-1 Case B-2 Case B-3

Natural Fracture Length (m)

50 26 or 74 26 or 74 38, 74, 4, 62, 54, 18, 90, 60

51

Fig 5.14 shows a schematic of each model set-up including the new base case. In Case B-1, two

sets of natural fractures with different lengths are placed alternatively. Case B-2 consists of four

evenly-distributed short natural fractures on one side and four long ones on the other side. Case

B-3 shows a case where eight natural fractures with different lengths are randomly placed along

the bi-wings of the hydraulic fracture. The pressure distributions at the end of 4.5 years for all

four cases are shown in Fig 5.15.

Figure 5.14 Schematic of non-symmetric drainage for the Bakken well

(a) – New Base Case; (b) Case B-1; (c) – Case B-2; (d) – Case B-3

(a) (b)

(c) (d)

52

Figure 5.15 Pressure (in kPa) distribution of the Bakken well after 4.5 years for

(a) – New Base Case; (b) Case B-1; (c) – Case B-2; (d) – Case B-3

Fig 5.16 compares the production performances of the three cases with that of new base case,

and the differences at early times are not noticeable. Once again, at early times, flow contribution

from natural fractures and matrix are not important. When comparing the production

performance from 100 days to 4.5 years, the effects of natural fractures become more apparent.

All three cases of B-1 to B-3 have the same average natural fracture length, and the resultant

(a) (b)

(c) (d)

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1,693

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28,827

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46,916


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53

total area available for flow between natural fracture and matrix would also be nearly identical.

The minor observable difference would most likely due to the heterogeneous distribution of

natural fracture lengths. It can be noted that oil rate for Case B-3 is slightly lower than those for

other cases. That is because arrangement of natural fractures in Case B-3 does not allow even

symmetric drainage from both sides of the hydraulic fracture. Pressure depletion and ensuing

drainage are less effective on the side of the horizontal well with less contact area between

natural fractures and matrix. The new base scenario gives the highest production rate because

natural fractures are evenly distributed, allowing more effective drainage across the entire

domain. All cases would converge again in long-term production (shown in Fig 5.16c) when

transient flow in matrix begins. Similar to the observations in section 3.1, the results presented in

this section would imply that the assumption of symmetric drainage in analytical solutions may

lead to underestimation of fracture half-length and the associated SRV.

(a) (b)

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)New Base Case Case B-1Case B-2 Case B-3

0.1

1

100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)New Base Case Case B-1Case B-2 Case B-3

54

(c)

Figure 5.16 Log-log plot of daily oil rate as a function of time of the Bakken well for Case B

(a) – early-time istory for 4.5 years; (b) – mid-time history between 100 to 10000 days; (c) – entire history for 20

years.

5.5 Heterogeneous Fracture Properties

The effect of fracture connectivity on rate transient response is studied next. Another assumption

which plays an important role in analytical models is that natural fractures are fully connected

between hydraulic fracture stages. That means for a fixed number of hydraulic fracturing stages

with uniform spacing, the natural fractures are connected across all stages (i.e., df = LF in Fig.3).

The ratio of natural fracture length df to hydraulic fracture spacing LF is defined as Rl, which

reflects the degree of natural fracture connectivity. As shown in Fig 5.17, if Rl equals to zero, the

model is equivalent to a dual-porosity model. On the other hand, if Rl equals to one, the model

has fully-connected natural fractures. Models with different values of Rl (0, 0.1, 0.3, 0.5, 0.7 and

1.0) are constructed by reducing the value of df, in order to study the relationship between natural

fractures connectivity and rate transient response for the Bakken well.

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

New Base Case Case B-1 Case B-2 Case B-3

55

Figure 5.17 Schematic of fully-conncected and non-conncected natural fractures for the Bakken well

(a) – Rl=1.0; (b) – Rl=0.5; (c) – Rl=0.25; (d) – Rl=0

It is evident in Fig 5.18 that as the value of Rl increases, oil production would also increase,

especially during later time when the drainage is dominated by linear flow in natural fractures

and bilinear flow in matrix and natural fractures. This observation implies that the assumption of

fully-connected natural fractures in the analytical model may underestimate nf.

(a) (b)

(c) (d)

56

Figure 5.18 Log-log plot of daily oil rate as a function of time of the Bakken well for cases with varying natural

fracture connectivity (4.5 years)

Another assumption of analytical models is that hydraulic fractures between two parallel

horizontal wells are fully-connected. That means the distance between two parallel horizontal

wells is the same as length of one hydraulic fracture (i.e., 2xf = Lw). The ratio of hydraulic

fracture length 2xf to distance between two parallel horizontal wells Lw is defined here as RL.

Several schematics with different values of RL are shown in Fig 5.19. RL = 1 represents a dual-

porosity system, while RL = 0 represents a single porosity system.

0.1

1

10

1 10 100 1000 10000

Oil

Rat

e (

m3

/day

)

Time (day)

Rl=0 Rl=0.1

Rl=0.3 Rl=0.5

Rl=0.7 Rl=1.0

Rl (df/LF)

increasing from

zero to one

57

Figure 5.19 Schematic of fully-conncected and non-conncected hydraulic fractures for the Bakken well

(a) – RL=1.0; (b) – RL=0.75; (c) – RL=0.5; (d) – RL=0.25

Fig 5.20 shows that as the value of RL increases, oil production would also increase.

Interestingly, when hydraulic fracture length is very short (RL = 0.224), the signatures and slopes

of production plot are very different from dual-porosity (RL = 1.0) plot. The slope is expected to

change from a half slope to a quarter slope and finally to a half slope in a normal dual-porosity

system (Fig 5.20). This sequence of slope change represents linear, bilinear and linear flow in

fracture, matrix-fracture and matrix, respectively. However, when RL equals to 0.224, we observe

a change from unit slope to half slope. The occurrence of early unit-slope and the absence of

(a) (b)

(c) (d)

58

early half- and quarter-slope indicate that the transient flow in relatively small hydraulic

fractures is masked by pseudo steady-state pressure depletion. In other words, the transient

behavior has quickly reached the tip of short hydraulic fractures.

Figure 5.20 Log-log plot of daily oil rate as a function of time of the Bakken well for cases with varying hydraulic

fracture connectivity (4.5years)

Furthermore, pressure map shown in Fig 5.21 indicates that there is pressure drop at the tips of

hydraulic fractures. This observation could explain why signatures of rate-time plot for relatively

0.01

0.1

1

10

1 10 100 1000 10000

OIl

Rat

e (

m3

/day

)

Time (day)

RL=0.224 RL=0.515

RL=0.7575 RL=1

RL increasing

from 0.224 to

1.0

0.01

0.1

1

10

1 10 100 1000 10000

OIl

Rat

e (

m3

/day

)

Time (day)

RL=0.224 RL=1

59

small RL would be different from those with high RL value. It also implies that the occurrence of

early unit slope could be an indication of limited hydraulic fracture penetration. Similar to the

effects of fully-connected natural fractures, the assumption of fully-connected hydraulic fracture

may underestimate the distance between two horizontal wells, which could potentially impact

future development design such as optimal well spacing.

Figure 5.21 Pressure (in kPa) distribution of the Bakken well after 4.5 years when RL=0.224

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File: 0.224 new.irfUser: Min YueDate: 2014/3/5

Scale: 1:376.160476 Y/X: 1.00:1Axis Units: m

1,191

5,763

10,336

14,908

19,481

24,053

28,626

33,198

37,771

42,344

46,916


60

6 Influence of Multi-phase Effects on oil Production

6.1 The Role of Gas Dissolution

Gas dissolution and multiphase flow effects could potentially contribute to early decline in

production, as often observed in many tight oil wells. However, these effects are typically

ignored in most analytical models. To understand how solution gas may influence two-phase oil-

gas flow, a model based on the Bakken well base case is considered. In order to provide a more

detailed description of the matrix-fracture interface, of the model size has been reduced to 10.6m

× 20.5m. The model is initialized with no free gas and zero water saturation. The relative

permeability and capillary pressure relationships for matrix and fracture system are assigned

according to the multiphase flow functions presented in Gdanski and Fulton (2009). Zero

capillary pressure and linear relative permeability functions (stick curves with exponent = 1.0)

are assigned in the fractures (Papatzacos and Skjæveland, 2002). The effects of gas adsorption

are ignored in this work, since Wu et al. (2014) estimated the contribution of gas

adsorption/desorption to the total gas production to be approximately 13%. Gas saturation profile

in the hydraulic fracture is shown in Fig 6.1. The initial increase and decrease in gas saturation

during the first few days may be attributed to the large disparity between initial reservoir

pressure and bottom-hole pressure; pressure in the hydraulic fracture drops suddenly, and a

significant amount of solution gas has evolved and is produced immediately.

61

Figure 6.1 Average gas saturation in hydraulic fracture during the first 30 days for the Bakken well

Next, a set of triple-porosity model with two natural fractures placed evenly on each side of the

horizontal well is constructed to assess the impacts of the amount of gas dissolution and

multiphase flow functions on production performance. The relative permeability and capillary

pressure functions in the natural fracture is assigned according to Iwai (1976) and Killough

(1976). The amount of solution gas is varied with different bubble-point pressure (Pb). The

bottom-hole flowing pressure, Pwf, is kept constant to ensure that variation in production is

attributed to solution gas alone. The production profiles and input PVT data for Pb = 10 and 35

MPa are compared in Fig 6.2 and Fig 6.3. In order to compare the combined energy content

obtained from the produced oil and gas, the total hydrocarbon rate based on the approximate

energy released by burning one barrel of oil is computed (British Petroleum, 2011).

62

(a) (b)

(c)

Figure 6.2 Input PVT data for the Bakken models with different bubble-point pressure

(a) – oil formation volume factor; (b) – solution gas-oil ratio; (c) – gas expansion factor.

(a) (b)

1

1.1

1.2

1.3

1.4

0 20000 40000 60000

Bo

PPb=10000 Pb=35000

1

21

41

61

81

101

121

0 20000 40000 60000

Rs

PPb=10000 Pb=35000

1

1.1

1.2

1.3

1.4

0 20000 40000 60000

Bo

PPb=10000 Pb=35000

1

21

41

61

81

101

121

0 20000 40000 60000

Rs

PPb=10000 Pb=35000

1

51

101

151

201

251

0 10000 20000 30000 40000

Eg

PPb=10000 Pb=35000

0.001

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1000

Oil

Rat

e (

m3

/day

)

Time (day)Pb=10000 Pb=35000

1

10

100

1000

10000

0.01 0.1 1 10 100 1000

Gas

Rat

e (

m3

/day

)

Time (day)Pb=10000 Pb=35000

63

(c)

Figure 6.3 Log-log plots of production profiles for the Bakken models with different bubble-point pressure

As Pb increases, both the oil rate and total hydrocarbon rate are reduced, while gas rate is

increased. This is because with the same initial pressure, higher Pb implies a larger pressure

range over which free gas would exist. Given that the mobility for the gas phase is higher than

that of the oil phase, oil production is reduced. Therefore, ignoring the presence of two phase oil-

gas flow in analytical models could lead to an overestimation of fracture parameters such as

effective fracture volume and fracture length.

The influences of relative permeability functions in the natural fracture are studied next. Fig 6.4

compares the production profiles for cases with different krog endpoint ranging from 0.1 to 1.0.

As the value of krog endpoint decreases, lower oil production can be achieved. The reasoning for

this decrease in production is two folds: decrease in hydraulic connection for the oil phase and

shifting the intersection of oil and gas relative permeability curves towards the higher liquid

saturation scale (i.e., there is a smaller saturation range over which the mobility of the oil phase

is higher than the gas phase).

0.1

1

10

100

1000

0.01 0.1 1 10 100

Tota

l Hyd

rob

arb

on

(b

bl/

day

)

Time (day)

Pb=10000 Pb=35000

64

(a) (b)

(c)

Figure 6.4 Log-log plots of production profiles for the Bakken models with different relative permeability

endpoints

Fig 6.5 compares the production profiles for cases with different gas relative permeability

exponent, a1, ranging from 1.0 to 3.0. Slightly lower oil and hydrocarbon production rates are

observed for the case with higher values of a1, especially at late time. It has already been

highlighted that ignoring the presence of two phase oil-gas flow could lead to an underestimation

of fracture parameters such as effective fracture volume and fracture length; the conclusions

derived from Figs 6.4-6.5 would imply that this underestimation would be more severe, if krog is

low and/or gas relative permeability exponent (a1) is high.

0.1

1

10

100

0.01 0.1 1 10 100

Oil

Rat

e (m

3/d

ay)

Time (day)Case C-1 Case C-2 Case C-3

Case C-4 Case C-5

Endpoint increases from 0.1 to 1.0

10

100

1000

10000

0.01 0.1 1 10 100

Gas

Rat

e (m

3/d

ay)


Case C-4 Case C-5

1

10

100

1000

0.01 0.1 1 10 100

Tota

l Hyd

rob

arb

on

(bb

l/d

ay)


Case C-4 Case C-5

0.1

1

10

100

0.01 0.1 1 10 100

Oil

Rat

e (m

3/d

ay)


Case C-4 Case C-5


10

100

1000

10000

0.01 0.1 1 10 100

Gas

Rat

e (m

3/d

ay)


Case C-4 Case C-5

1

10

100

1000

0.01 0.1 1 10 100

Tota

l Hyd

rob

arb

on

(bb

l/d

ay)


Case C-4 Case C-5

0.1

1

10

100

0.01 0.1 1 10 100

Oil

Rat

e (

m3

/day

)


Case C-4 Case C-5


10

100

1000

10000

0.01 0.1 1 10 100

Gas

Rat

e (

m3

/day

)


Case C-4 Case C-5

1

10

100

1000

0.01 0.1 1 10 100

Tota

l Hyd

rob

arb

on

(b

bl/

day

)


Case C-4 Case C-5

65

(a) (b)

(c)

Figure 6.5 Log-log plots of production profiles for the Bakken models with different relative permeability

curvatures

0.1

1

10

100

0.01 0.1 1 10 100

Oil

Rat

e (

m3

/day

)

Time (day)

a1=1 a1=2 a1=3

𝐾𝑟𝑓 = 𝐾0𝑟𝑓⬚

× 𝑆⬚𝑎1

10

100

1000

10000

0.01 0.1 1 10 100

Gas

Rat

e (

m3

/day

)

Time (day)a1=1 a1=2 a1=3

1

10

100

1000

0.01 0.1 1 10 100

Tota

l Hyd

rob

arb

on

(b

bl/

day

)

Time (day)

a1=1 a1=2 a1=3

66

6.2 The Role of Capillary Discontinuity

Unlike most experiments and analytical models that use coreflood to present capillary

discontinuity, we have models under reservoir conditions with a high capillary pressure

difference between fractures and matrix. Since hydraulic fracture has high conductivity, its

corresponding capillary pressure is significantly lower than that in the rock matrix (Aguilera,

1980). Therefore, zero capillary pressure and linear relative permeability functions are assigned

in the fractures (Papatzacos and Skjæveland, 2002). This zero capillary pressure in the hydraulic

fracture is to imitate the zero capillary pressure boundary condition in core-flooding

experiments. A section of the reservoir between two hydraulic fractures showing in Fig 4.1 is

modeled.

We use a typical correlation, Corey correlation, to represent relative permeability in the matrix

(Brooks and Corey, 1964).

Correlations of oil relative permeability can be defined as below:

oC

wnorowwrow SkSk 1 (5)

Where Swn is defined as a normalized water saturation value:

orwwi

wiwwwnwn

SS

SSSSS

1 (6)

Where Swi is irreducible water saturation; Sorw is residual oil saturation.

Co is empirical parameters obtained from measured data or history matching; orowk is called the

end point of oil relative permeability.

67

Capillary pressure in matrix is assigned according to the correlation in Eq. (7) (Gdanski et al.,

2009).

3

1

'

2

6.89476

a

c aw

Pka S

(7)

' is the interfacial tension of water and oil (30 dynes/cm). For the matrix, a measure of pore

structure a3 equals to 0.5 (Bradley, 1992). To represent low-permeability reservoirs, a1 and a2

equal to 1.86 and 6.42 respectively (Holditch, 1979).

Dual-porosity and triple-porosity models with one hydraulic fracture in the middle and

perpendicular to the horizontal wellbore were built. By using these models, influences of fracture

face-effect on production profile in fractured reservoir can be investigated. We use logarithmic

grid size distribution near fracture to capture flow behavior by using saturation profile. In this

part, a dual-porosity model called base case B is built. Since the irreducible water saturation is

0.2, the initial water saturation is to 0.5 to model mobile water and no free gas at initial

conditions. To ensure that only two flowing phases exist and that there is zero gas production,

the bubble-point pressure is set at 15000 kPa, in comparison with initial reservoir pressure of

48630 kPa. Fig 6.6 shows the simulated saturation versus distance to the fracture-matrix

interface for a dual-porosity model at different times.

68

(a) (b) (c)

Figure 6.6 Oil and water saturation profiles in matrix as a function of distance from the fracture-matrix interface

for the dual-porosity model at:

(a)- 10 days; (b)- 30 days; (c)- 90 days

The interface between fracture and matrix is located at x = 0. Oil saturation in matrix increases

from its initial value due to oil expansion with pressure decline. Highest water saturation is

observed at the interface, and as production continues, this highest value increases from its initial

value (0.5) as a result of enlarged water blockage at the interface. This water blockage is a

distinctive characteristic of fracture-face effect and may have influences on production.

Water imbibition due to capillary pressure during extended shut-in can lead to the transfer of oil

from matrix into fracture (Cheng, 2012). In this part, we try to investigate the relationship

between capillary pressure functions and fracture-face effect under reservoir conditions. We are

0.3

0.4

0.5

0.6

0.7

0 2 4 6

S

X (m)water oil

10 days

0.3

0.4

0.5

0.6

0.7

0 2 4 6

S

X (m)water oil

30 days

0.3

0.4

0.5

0.6

0.7

0 2 4 6

S

X (m)water oil

90 days

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6

S

X (m)water oil

0.5

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6

S

X (m)water oil

0.5

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6

S

X (m)water oil

0.5

40% of initial oil is produced

69

investigating the effect of matrix capillary pressure by running three dual-porosity models with

different capillary pressure functions as shown in Fig 6.7.

Figure 6.7 Different capillary pressure curve for matrix

We use three different capillary pressures curves to investigate the effects of fracture-face effect

at fracture-matrix interface. Here, we hypothesize that with higher Pc of matrix, capillary end-

effect becomes more pronounced and oil production rate drop. Higher Pc of matrix may cause

larger water blockage near the interface of matrix and fracture, and therefore, oil flow from

matrix to fracture can be hindered.

Fig 6.8 shows oil saturation in matrix close to frac-matrix interface (0.01025 m) in models with

different capillary pressure in the first day. It is shown in Fig 6.8 that after 1 hour of production,

oil saturation in matrix close to frac-matrix interface decreases. It is observed in Fig 6.8 that oil

saturation drop is sharper for the case with higher capillary pressure. And low oil saturation at

the interface (which means high water saturation) leads to water blockage around fracture-matrix

interface.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0 0.2 0.4 0.6 0.8 1

Pc

(kP

a)

Sw

Pc1 Pc2 Pc3

70

(a) (b)

(c) (d)

Figure 6.8 Oil and water saturation profiles in matrix as a function of time for the dual-porosity model with

different capillary pressure:

(a)- oil saturation in 1 day; (b)- oil saturation in 0.04 day

(c)- water saturation in 1 day; (b)-water saturation in 0.04 day

Saturation comparisons of models with different capillary pressure in matrix at 30 days are

plotted in Fig 6.9. Both oil saturation and water saturation versus distance to frac-matrix

interface are shown. It can be seen that higher capillary pressure in matrix causes higher water

saturation at interface. This higher water saturation leads to water blockage around the interface.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30

So

Time (hrs)

Higher Pc, lower oil saturation at interface

Lower Pc

Higher Pc

0.3

0.35

0.4

0.45

0.5

0.55

0 0.5 1 1.5

So

Time (hrs)

Lower Pc

Higher Pc

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30

Sw

Time (hrs)

Lower Pc

Higher Pc

0.4

0.45

0.5

0.55

0.6

0.65

0 0.5 1 1.5

Sw

Time (hrs)

Lower Pc

Higher Pc

71

(a) (b)

Figure 6.9 Oil and water saturation profiles in saturation in matrix as a function of distance from the fracture-

matrix interface for the dual-porosity models with different capillary pressure at 30 days:

(a)-oil saturation in 0.6 m; (b)-water saturation in 0.6 m

Fig 6.10 shows production comparison of models with various capillary pressure in matrix. Fig

6.10 (a) and Fig 6.10 (b) are oil rate and water rate during the whole production process. Both oil

rate and water rate decrease with increasing of capillary pressure in matrix after one day. As we

mentioned above, higher capillary pressure brings more water blockage around frac-matrix

interface and this water blockage prevents both oil and water in matrix from going into fracture.

An interesting thing to note is that in production profile shown Fig 6.10, oil rate increases as

capillary pressure increases while water rate has a sharp drop in the first day.

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6

So

X (m)Pc1 Pc2 Pc3

Pc1 < Pc2 < Pc3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6

Sw

X (m)

Pc1 Pc2 Pc3

72

(a)

(b)

Figure 6.10 Log-log plot of daily rate versus time for the dual-porosity models with different capillary pressure in

matrix during 91 days:

(a)-oil rate from very beginning; (b)- water rate from very beginning

Both oil rate and water rate decrease with increasing of capillary pressure in matrix after one day.

As we mentioned above, higher capillary pressure brings more water blockage around frac-

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)

Pc1 Pc2 Pc3

Pc1<Pc2<Pc3

0.1

1

10

100

0.00001 0.001 0.1 10

Wat

er

Rat

e (

m3

/day

)

Time (day)

Pc1 Pc2 Pc3

73

matrix interface and this water blockage prevents both oil and water in matrix from going into

fracture. An interesting thing to note is that in production profile shown in Fig 6.10 (a) and Fig

6.10 (b), oil rate increases as capillary pressure increases in the first day. This increasing of oil

rate can be explained by Fig 6.11. Fig 6.11 shows average oil saturation in fracture by using time

scale.

(a) (b)

Figure 6.11 Average oil saturation in fracture in models in as a function of time for the dual-porosity model

different capillary pressure:

(a)- in 112 days; (b)- at 1 hour

We observe in Fig 6.11 that oil saturation in fracture is higher for the model with higher capillary

pressure in matrix, after 1 hour of production. With higher capillary pressure in matrix, it is

harder for both oil and water in matrix to flow into fracture. In the beginning, with higher Pc, the

difference of oil phase pressure between in the nearby matrix and the fracture is higher, leading

to higher oil drainage in the matrix. Therefore, higher oil rate and lower water rate can be

achieved. As production continues, fracture-face effect becomes more important, and water

blockage is slowing down both oil and water flows.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150

So

Time (day)

Higher Pc, higher oil saturation in fracture, especially at very beginning.

Higher Pc

Lower Pc

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5

So

Time (hrs)

Higher Pc

Lower Pc

74

Fig 6.12 compares cumulative oil of models with various capillary pressure models in matrix. It

is easy to see that although capillary pressure changed, cumulative oil production does not

change much. In Wang and Leung’s (2015) paper, quite similar results are observed in soaking

models.

Figure 6.12 Log-log plot of cumulative oil versus time for the dual-porosity models with different capillary

pressure in matrix during 91 days

The bold horizontal line represents initial oil volume in fracture at initial conditions. Based on

Fig 6.12 and Fig 6.10 (a), we hypothesize that most of produced oil comes from fracture at

beginning (less than 3 hrs). However, cumulative oil in 3 hours is beyond initial oil volume in

fracture. This is because some oil in matrix goes into fracture due to the draw down at early

times.

0.0001

0.001

0.01

0.1

1

10

100

1000

0.00001 0.001 0.1 10 1000

Cu

mu

lati

ve O

il (m

3)

Time (day)

Pc1 Pc2 Pc3 Qo in frac

Pc1<Pc2<Pc3

Initial oil volume in fracture at beginning

75

Another set of models with different initial water saturation in matrix are constructed to

investigate influences of initial water saturation on production under different capillary pressure.

Results of oil rate with different capillary pressure in models with three different initial water

saturation levels are shown in Fig 6.13.

(a) (b)

(c)



Initial water saturation in matrix equals to (a)-0.3; (b)- 0.5; (c)-0.7

We observe that oil rate decreases with increasing initial water saturation in matrix. Here, we

define Pcq to be the difference in oil rate between models with lower and higher capillary

0.1

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)

Pc1

Pc3

0.1

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)

Pc1

Pc3

0.1

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)

Pc1

Pc3

76

pressure in matrix. In Fig 6.13, Pcq increases with increasing initial water saturation. We

compute Pcq as a function of time. When initial water saturation reaches to 0.7, its average

differential ( 1

/PcPc

qq 100%) value over the entire production period is around 30%. This is

because with higher initial water saturation in matrix, there is more mobile water in matrix. More

mobile water in matrix provides more favorable conditions for the occurrence of fracture-face

effect. This leads to more water blockage around interface which decreases oil production.

After discussing impacts of capillary pressure with dual-porosity models, triple-porosity models

with one hydraulic fracture and eight secondary fractures are constructed. Relative permeability

functions in secondary fracture are adopted from Aguilera et al. (1980). And a capillary function

for secondary system is assigned from Eq. (7) before. We expected with more frac-matrix

interfaces in triple-porosity model, fracture-face effect became more obviously. Fig 6.14 is water

and oil saturation profiles in matrix as a function of distance from the hydraulic fracture interface

(X) and secondary (micro) fracture interface (Y) at different times.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

S

X (m)water oil

5 days

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

S

X (m)water oil

25 days

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

S

X (m)water oil

50 days

77

(a)-1 (a)-2 (a)-3

(b)-1 (b)-2 (b)-3

Figure 6.14 Oil and water saturation profiles in matrix as a function of

(a) distance from the hydraulic fracture-matrix interface X and (b) distance from the secondary fracture-matrix

interface Y for the triple-porosity model at

(1): 5 days; (2); 25 days; and (3) 50 days.

Fig 6.14 (a) shows the evolution of saturation profiles in matrix as a function of distance from

the hydraulic fracture-matrix interface with time. The region near the interface is enlarged to

illustrate the fracture-face effect more clearly. Similarly, Fig 6.14 (b) presents the saturation

profiles matrix as a function of distance from the hydraulic fracture-matrix interface at different

time. It is shown in both Fig 6.14 (a) and Fig 6.14 (b) that as time goes by, water saturation close

to frac-matrix interface increases. It is discovered that fracture-face effect appears in two

different systems (hydraulic fracture-matrix, secondary fracture-matrix) in triple-porosity models

(Fig 6.14) due to high Pc difference between matrix and fracture.

The effects of capillary pressure functions in matrix on water saturation and on oil production are

shown in Fig 6.15. Three triple-porosity models with varying capillary pressure in matrix are

constructed.

0

0.2

0.4

0.6

0.8

1

0 1 2 3

S

Y (m)water oil

5 days

0

0.2

0.4

0.6

0.8

1

0 1 2 3

S

Y (m)water oil

25 days

0

0.2

0.4

0.6

0.8

1

0 1 2 3

S

Y (m)water oil

50 days

78

(a) (b)

(c) (d)

Figure 6.15 Triple-porosity models with different capillary pressure:

(a)- Log-log plot of daily oil rate versus time during 91 days; (b)- Water saturation of HF-M system at 30 days;

(c)- Water saturation of SF-M system at 30 days; (d)- Water saturation of SF-HF system at 30 days.

(Values of Pc1, Pc2 and Pc3 are shown in Fig 6.7)

With increasing capillary pressure in matrix, water saturation in matrix close to frac-matrix

interface increases, while oil rate decreases. These observations are quite similar to the results in

dual-porosity model. However, the difference in oil rate shown in Fig 6.15 (a) is not pronounced.

Table 6.1 shows a comparison between dual-porosity and triple-porosity models on water

saturation at the interface in HF-M system and SF-M system.

Table 6.1 Water saturation at interface in dual-porosity and triple-porosity models with different capillary

pressure (Pc1 and Pc3)

0.1

1

10

100

0.00001 0.001 0.1 10 1000

Oil

Rat

e (

m3

/day

)

Time (day)Pc1 Pc2 Pc3

Pc1<Pc2<Pc3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6

Sw

X (m)

Pc1 Pc2 Pc3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6

Sw

Y (m)Pc1 Pc2 Pc3

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6

Sw

X (m)

Pc1 Pc2 Pc3

79

Swinter Swinter (Pc=Pc1) Swinter (Pc=Pc3) ∆ Swinter

Dual-porosity (HF-M)

0.6181 0.8445 0.2264

Triple-porosity (HF-M)

0.6305 0.8479 0.2174

Triple-porosity (SF-M)

0.7388 0.8641 0.1253

Compared with dual-porosity model in Table 6.1, triple-porosity model has higher water

saturation in both HF-M and SF-M systems. This means fracture-face effect becomes more

pronounced in models with more interfaces. However, the difference in oil production ( Pcq )

that we compute as a function of time in triple-porosity model does not increase compared with

Pcq in dual-porosity models. The contrast between different porosity systems in a medium

consisting of matrix, secondary fractures and hydraulic fracture is less in comparison to a

medium consisting of matrix and hydraulic fracture only. Fracture-face effect has fewer

influences on oil rate in triple-porosity models compared with impacts on oil rate in dual-

porosity models shown in Fig 6.10 (a).Since the fracture-face effect is more pronounced between

HF and M, the dual-porosity model is employed in the next section to study the impacts on

matrix relative permeability functions.

Next, we investigate possible relationships between relative permeability and production

performance. Two models with different values of relative permeability endpoint and curvature

in matrix are tested

According to Eq. (5) and Eq. (6) shown before, two parameters determine value of relative

permeability, orowk and oC . By changing these two parameters, sensitivity of relative

permeability value on production are assessing. The first parameter tested here is value of orowk .

80

This value we call it endpoint of rowk in matrix and it changes from 0.4 to 0.9, as shown in Fig

6.16.

Figure 6.16 Different relative permeability in matrix system by changing endpoint of krow

Fig 6.16 shows that as the endpoint of krow increases, the value of oil relative permeability in

matrix increases. Fig 6.17 presents the oil and water saturation profiles in matrix as a function of

distance from the HF-M interface by using models with different endpoint of krow. It is observed

that as the endpoint of krow increases, oil saturation decreases while water saturation increases.

According to Fig 6.16, the intersection point of oil and water relative permeability curves goes to

right, as the endpoint of krow increases. This observation means that matrix becomes more water-

wet and capillary end effect becomes more pronounced.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

krow

krw

0.9

0.7

0.4

81

(a) (b)

Figure 6.17 Saturation profiles in saturation in matrix as a function of distance from the fracture-matrix

interface with different endpoint of krow at 30 days:

(a)-oil saturation; (b)- water saturation

Fig 6.18 shows the results of log-log plot of daily production versus time during 112 days using

models with different krow endpoints in matrix.

(a)

Endpoint of krow

Increases

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)

maxkrow=0.4 maxkrow=0.7 maxkrow=0.9

Endpoint of

krow increases

82

(b)

Figure 6.18 Log-log plot of daily rate versus time for the dual-porosity models with different kro endpoints in


(a)-oil rate; (b)- water rate

Oil rate increases as oil relative permeability increases. On the other hand, water rate decreases

by increasing endpoint of krow. Although water saturation at interface is larger which means more

water blockage around interface as endpoint of krow increases, higher krow endpoint gives higher

value of krow at interface water saturation. This leads to higher oil transmissibility and lower

water transmissibility.

The second sensitivity testing is curvature of oil relative permeability in matrix. This curvature is

Co in equation (1) which influences relative permeability directly. When this exponent equals to

1.0, relative permeability curves become straight line. Exponent of oil relative permeability

curvature in matrix defined as Co changes from 2.0 to 5.0 which is shown in Fig 6.19.

0.1

1

10

100

0.00001 0.001 0.1 10

Wat

er

Rat

e (

m3

/day

)

Time (day)

maxkrow=0.4 maxkrow=0.7 maxkrow=0.9

83

Figure 6.19 Different relative permeability in matrix system by changing curvature of krow

We hypothesize that due to increasing Co, changing of water saturation have stronger impacts on

oil relative permeability. It will result in fracture-face effect increasing and leads to oil rate

reduction. Fig 6.20 is oil and water saturation in matrix close to fracture. It shows that when Co

in matrix is higher, more water blockage appears near fracture which means fracture-face effect

becomes more pronounced, and it hinders oil transport from matrix to fracture.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Kr

Sw

krow

krw

84

(a) (b)

Figure 6.20 Saturation profiles in saturation in matrix as a function of distance from the fracture-matrix

interface with different curvature of krow at 30 days:

(a)-oil saturation; (b)- water saturation

Similar to the observation in Fig 6.6 for the base case, oil saturation in matrix increases due to oil

expansion with reservoir pressure decline during production. Fig 6.21 is log-log plot of daily

production versus time during 112 days. Oil rate shown in Fig 6.21 decreases while water rate

increases with Co increasing. This observation is consistent with the characteristics of oil

saturation profiles shown in Fig 6.20. With Co increasing, more oil remains in the matrix due to

lower krow at a given saturation; therefore, oil rate decreases.

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6

So

X (m)Co=2 Co=3 Co=5

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6

Sw

X (m)

Co=2 Co=3 Co=5

Curvature of krow decreases

85

(a)

(b)

Figure 6.21 Log-log plot of daily rate versus time for the dual-porosity models with different krow curvature in


(a)-oil rate; (b)- water rate

To investigate how much capillary pressure and relative permeability influence production. We

do an additional sensitivity. Here, we do a combination sensitivity with capillary pressure and

relative permeability curvature. Fig 6.22 is a schematic of the whole procedure. Firstly, Pc1 and

0.1

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)

Co=2 Co=3 Co=5

0.1

1

10

100

0.00001 0.001 0.1 10

Wat

er

Rat

e (

m3

/day

)

Time (day)

Co=2 Co=3 Co=5

Curvature increases

86

Pc3 shown in Fig 6.7 are used in Matrix. Water saturation profile at particular day can be

obtained by using models with different capillary pressure in matrix. Different value of Sw close

to frac-matrix interface (0.01025 m) in models with Pc1 and Pc3 can be read in water saturation

profiles (Sw1 and Sw3 in Fig 6.22 (b)). According to Fig 6.16, when endpoint of krow equals to 0.9,

rowk can be obtained. Then this whole procedure is repeated by using endpoint of krow equals to

0.4. Fig 6.23 is saturation profile at 30 days by using Pc1 and Pc3 on matrix with different

relative permeability curvature. We use these plots to obtain water saturation at interface.

(a)- (1) (b)- (1) (c)- (1)

(a)- (2) (b) - (2) (c)- (2)

Figure 6.22 Schematic of calculation procedure for ∆krow:

(1)- endpoint of krow=0.9; (2)- endpoint of krow=0.4

(a)-different Pc in matrix; (b)- water saturation profiles of models with different Pc in matrix;

(c)-oil relative permeability

cP wS

wSwS( )X m

( )kPa1cP

3cP

1wS

3wS

3wS1wS

Δ

0.9

cP wS

wSwS( )X m

( )kPa1cP

3cP

1wS

3wS

3wS1wS

Δ

0.4

87

(a)

(b)

Figure 6.23 Saturation profiles in saturation as a function of distance from the fracture-matrix interface with

different capillary pressure in matrix at 30 days:

Endpoint of krow equals to (a)-0.4; (b)- 0.9

Take orowk =0.4 as an example:

According to Fig 6.23 (a), wS =0.8750-0.6425=0.2325;

According to Fig 6.16, 2rowk = krow(0.6425)- krow(0.8750) =0.02

Table 6.2 is list of rowk . We observe that with increasing the endpoint of oil relative

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6

S

X (m)

So/Pc1 So/Pc3 Sw/Pc1 Sw/Pc3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6

S

X (m)

So/Pc1 So/Pc3 Sw/Pc1 Sw/Pc3

88

permeability, rowk increases. This result demonstrates that Pcq decreases as oil relative

permeability endpoint increases.

Table 6.2 Value of rowk in conditions of different relative permeability curvature

Endpoint of krow =0.4 Endpoint of krow =0.9

rowk 0.02 0.07

Log-log plots of daily rate versus time with different capillary pressure and relative permeability

in matrix are shown in Fig 6.24. Results show that with increasing of krow endpoint, oil rate

decreases.

(a) (b)



endpoint of krow equals to (a)-0.4; (b)- 0.9

Using data shown in Fig 6.23, we calculate Pcq as a function of time in Fig 6.24. It is obvious

that Pcq increases when endpoint of oil relative permeability increases at the first 50 days. Due

to the endpoint increasing, rowk increases and this leads to Pcq increasing. This is consistent

with what we expected. After 50 days, due to higher oil rate in model with higher krow endpoint,

0.1

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)Pc1 Pc3

0.1

1

10

100

0.00001 0.001 0.1 10

Oil

Rat

e (

m3

/day

)

Time (day)Pc1 Pc3

89

drainage reaches boundary and Pcq goes to flat.

Figure 6.25 Δq as a function of time in models with different endpoint of krow

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20 40 60 80 100

∆q(m

3/day)

Time (day)maxkrow=0.4 maxkrow=0.9

90

7 Conclusions and Future Work

7.1 Conclusions

A workflow that integrates numerical simulation results with analytical solutions is implemented

to quantify the uncertainty in production performance predictions. Assuming that results

obtained from RTA would characterize the mean estimate of the corresponding fracture

parameters, additional heterogeneous models of fracture properties are subjected to flow

simulations to demonstrate their impact on production performance. Our results highlight the

non-uniqueness of fracture characterization in production data analysis. It is illustrated that

ignoring the uncertainty in history-matched parameters and assumptions associated with

analytical models could potentially over- or under-estimate production by up to 30%.

Certain assumptions in analytical models may not necessarily hold. Numerical models are

implemented in this study to assess their limitations and impacts. It is demonstrated that

analytical sequential-flow model underestimates drainage with time; therefore, history-matching

with analytical models alone would tend to overestimate fracture parameters including natural

fracture intensity. The assumption of uniformly-spaced fracture stages, however, could lead to

underestimation of hydraulic fracture half-length and the associated stimulated reservoir volume.

This could potentially impact future field development decisions such as placement of nearby

wells to maximize drainage.

Lengths and spacing between natural fractures play an important role, particularly at mid to late

times: higher production and more efficient drainage can be observed when natural fractures are

evenly distributed for a given number of natural fractures, especially when matrix permeability is

ultra-low. This also implies that the assumption of evenly-distributed natural fractures with

91

homogeneous properties could lead to underestimation of fracture half-length and the associated

stimulated reservoir volume.

A series of simulation models depicting different boundary conditions are also used to evaluate

the uncertainties due to pressure interference between natural fractures and inter-well fracture

communication. The connectivity of both natural and hydraulic fractures has a pronounced effect

on the rate transient behavior. Some of the flow regimes such as fracture linear transient flow

disappear as the secondary fractures between hydraulic fractures or hydraulic fractures between

lateral wells become disconnected. Oil production decreases more slowly during late time when

there is linear flow in natural fractures and bilinear flow in natural fractures and matrix. On the

other hand, disconnected short hydraulic fractures may lead to unit slope instead of half slope

and quarter slope at early time. The occurrence of early unit-slope and the absence of early half-

and quarter-slope indicate that the transient flow in relatively small hydraulic fractures is masked

by pseudo steady-state pressure depletion due to limited hydraulic fracture penetration. This may

contribute to additional uncertainties in fracture characterization with analytical models.

The approach presented in this study can be used as an avenue to quantify the effects of model

uncertainties on production performance prediction, and the method can be readily extended to

shale reservoirs. Given the subject of resource/reserve estimation of multi-fractured horizontal

wells in unconventional reservoirs remains challenging, this work highlights the importance of

adopting a modeling framework that takes into consideration the uncertainties in model

parameters when carrying out long-term production forecast.

Ignoring the presence of two phase oil-gas flow or assuming the absence of solution gas in tight

oil reservoirs could lead to overestimation of fracture parameters such as effective fracture

92

volume and fracture length and permeability. In particular, this overestimation would be more

severe, if krog is low and/or gas relative permeability exponent (a1) is high.

Capillary end-effects do exist at facture-matrix interface and cause water blockage at this

interface. This is similar to the end-effect observed in laboratory coreflooding tests. This

capillary discontinuity plays an unignorably role in matrix-fracture flow communication, which

in turn, influences water hydrocarbon production. Results show that less water blockage and

higher production are observed in cases with lower capillary contrast between fracture and

matrix. With higher capillary contrast between fracture and matrix, effects of imbibition become

more pronounced and more oil flows into fracture from matrix. However, cumulative oil

production does not change much. Also, Capillary end- effect has less influences on oil rate in

triple-porosity models compared with impacts on oil rate in dual-porosity models.

It is also observed that initial oil production increases slightly due to the displacement of oil by

water (wetting phase) near the fracture-matrix interface. This increase is more pronounced when

we increase the capillary pressure or initial water saturation in matrix blocks. However, after this

slightly increasing, whole production procedure is hampered as a result of increased water

blockage.

Relative permeability effects capillary discontinuity directly. Increasing oil relative permeability

endpoint or decreasing oil relative permeability curvature can lead to capillary end-effect more

pronounced. Models with different capillary pressure in matrix can have higher oil rate

difference as endpoint of oil relative permeability increases.

The results can be applied 1) for a more representative simulation of fractured horizontal wells

and 2) for designing efficient treatment technologies for the removal of water blockage which is

one of the key mechanisms for the production decline observed in the field.

93

7.2 Future Work

For future multi-phase production from fractured horizontal wells studies, the following future

research is recommended:

Both dual-porosity models and triple-porosity models in tight oil reservoir with three-

phase flow can be constructed to figure out three-phase effects on oil production,

especially long-term production.

The presented triple-porosity models in this thesis will be evaluated using more field data

from different formations and wells to do production prediction.

Stochastically distributed discrete fracture networks (DFN) can be employed in future

work. Branched natural (secondary) fracture with different length and position will be

modeled randomly. This can capture the uncertainties associated with reservoir

heterogeneity better.

An injector can be used in triple-porosity model in this thesis to simulate flow-back and

shut-in operation. Effects of capillary discontinuity from fractured horizontal well on

hydrocarbon production will be investigated using models with injector.

94

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